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.founded ow the: ordinal treatise 

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*•' } '• »• V « fcl * •• 1 V ; 

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Bureau of Reclamation 
Washington Office, Engineering Piles. 



'OF 

RETAINING WALLS. 


BY 

Professor WILLIAM CAIN, A.M., C.E. 

«t 

CAROLINA MILITARY ACADEMY. 

BOUNDED ON THE ORIGINAL TREATISE 
OF ARTHUR JACOB, C.E.) 


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to 







1° PREFACE. 


In attempting a revision of Jacob’s 
“ Retaining-Walls,” it was soon found that 
the entire treatise would have to be rewritten 
to bring it fully abreast with the times. 
The old theory that assumed the direction 
of the earth-thrust as normal to the inner 
face of the wall, or as having in all cases 
a direction parallel do the top surface, has 
been utterly expjoclbd by every experiment 
that has been performed ; and it is hoped 
that the time lias arrived for its permanent 
retirement. 

This work is divided into an Introduction, 
where this direction of the earth-thrust 
receives careful attention, and four follow¬ 
ing chapters, pertaining to reservoir-walls 
and the theory of retaining-walls, developed 
respectively by the graphical method, the 
analytical method, and finally by the experi- 

iii 



IV 


mental method, leading up, after the dis¬ 
cussion of all experiments available, to 
“the practical designing of retaining-walls.” 

In the brief discussion of dams, the 

/ 

occasion is taken to develop certain well- 
known elementary principles that are com¬ 
mon to retaining-walls as well as dams. 
In subsequent chapters of this work a good 
deal of new matter is given for the first 
time ; notabty in the analytical theory of 
the retaining-wall, and in the graphical dis¬ 
cussion of “ the limiting plane ” in Chap. II. 
The theory of the retaining-wall has been 
deduced, with the one assumption of a 
plane surface of rupture, from well-known 
mechanical laws; Coulomb’s “wedge of 
maximum thrust ” being incidentally proved 
in the course of the demonstration, but not 
assumed as a first principle. 

It is hoped that constructors will hail 
with delight the numerous experiments of 
Leygue and others, discussed in the fourth 
chapter, which lead to two semi-empirical 
methods which can be confidently used in 
practice for the design of retaining-walls. 
The practical tables given in this chapter 




V 


are for the first time published in this coun¬ 
try, and they are believed to be of great 
value. 

The aim of the author has been to prepare 
a treatise which should serve, at the same 
time, as a text-book for our engineering 
colleges, and as a manual for the practical 
engineer; and he trusts that he may have 
succeeded passably well in carrying it out. 

Wm. Cain. 

Charleston, S.C., May, 1888. 






TABLE OF CONTENTS. 


PAGE 

INTRODUCTION.1 

CHAPTER I. 

Reservoir-Walls.17 


CHAPTER II. 

Theory of Retaining-Walls.— Graph¬ 
ical Method.. 84 

CHAPTER III. 

Theory of Retaining-Walls. — Ana¬ 
lytical Method. 78 

CHAPTER IV. 

Experimental Methods. Comparison 
with Theory. The Practical De¬ 
signing of Retaining-Walls .... Ill 


APPENDIX. 

Design for a very high Masonry Dam, 1G1 

vii 











































































Bureau of Reclamation 
Washington Office, Engineering Piles. 


PRACTICAL DESIGNING 

OP 

RETAINING-WALLS. 


INTRODUCTION. 

1. The retaining or revetment wall is 
generally a wall of masonry, intended to 
support the pressure of a mass of earth or 
other material possessing some frictional 
stability. In certain cases, however, as in 
dock-walls, the backing or filling — as the 
material behind the wall is called — is liable 
to become in part or wholly saturated with 
water, so that the subject of water-pressure 
has to be considered to complete the inves¬ 
tigation. In cases where the filling is de- 
posited behind the wall after it is built, the 
full pressure due to the pulverulent fresh 
earth or other backing is experienced; and 
the wall is designed to meet such pressure, 
with a certain factor of safety, as near as it 




2 


can be ascertained. In time the earth 
becomes more or less consolidated by the 
settling due to gravity, vibrations, and rains, 
from the compressibility of the material, 
which thus brings into action those cohesive 
and chemical affinities which manufacture 
solid clays out of loosely aggregated mate¬ 
rials, and often causes the bank eventually 
even to shrink away from the wall intended 
to support it, when, of course, there will be 
no pressure exerted against the wall. 

2. Where a wall is built to support the 
face of a cutting, the pressure may be 
nothing at first, but it would be very unwise 
to make the wall much thinner than in the 
preceding case ; for it is a well-known fact 
of observation, that incessant rains often 
saturate the ground of open cuttings to 
such an extent as to bring down masses of 
earth, whose surface of rupture is curved, 
being more or less vertical at the top and 
approaching a cycloid somewhat in section ; 
the surface of sliding being so lubricated 
by the water that the pressure exerted hori¬ 
zontally by this sliding mass is even greater 
than for dry pulverulent materials. It is, 


3 


in fact, on this account, as well as from the 
force exerted by water in freezing, and from 
the disturbing influences caused by the 
passage of heavy trains, wagons, etc., which 
set up vibrations that lower the co-efficient 
of friction of the earth, and besides add 
considerably by their weight to the thrust 
of the backing, that a factor of safety 
against overturning and sliding of the wall 

O o o 

is introduced, which factor in practice gen¬ 
erally varies between two and three when 
the actual lateral pressure of the earth is 
considered. 

3. It is stated that retaining-walls in 
Canada require a greater thickness at the 
top to resist the action of frost than farther 
south where the frost does not penetrate the 
ground to so great a depth. Again, if the 
strata in a cutting dip towards the wall, 
with thin beds of clay, etc., interposed that 
may act as lubricants when wet, the press¬ 
ure against the wall may become enormous ; 
or if fresh earth-filling is deposited upon an 
inclined surface of rock, or other impervious 
material that may become slippery when the 
water penetrates and accumulates at its sur- 


4 


face, the pressure may become much greater 
than that clue to dry materials. It is found, 
too, that certain clays swell when exposed 
to the air with great force ; others, again, 
remain unchanged. In all such exceptional 
cases the engineer must use his best judg¬ 
ment after a careful study of the material 
he has to deal with. The theory and 
methods used in this book will not deal 
with such exceptional cases, but simply with 
dry or moist earth-filling supported by good 
masonry upon a firm foundation ; and it is 
believed the theory deduced will be of mate¬ 
rial assistance to any one who may have to 
deal with even very exceptional conditions, 
or, as in the case of military engineers, with 
the design of revetment-walls partly as a 
means of defence. 

4. When a retaining-wall fails, it is not 
generally from not having sufficient section 
for dry backing properly laid (in layers 
horizontal or inclined downwards from the 
wall), but because the earth has been dumped 
in any fashion against the wall, and no 
“weep holes” have been provided to let 
off the water that is sure in time of rains to 


5 


saturate the bank. If to this is added bad 
masonry, and a yielding foundation, or one 
liable to be washed out, the final destruc¬ 
tion of the wall can be pretty confidently 
counted on. 

5. The folio wing little table of weights 

O o 

and angles of repose of various materials 
used in construction may prove of assistance, 
but in any actual case the engineer should 
determine them by actual experiment: — 



Weight per Cubic 
Foot in Pounds. 

Angle of 
Repose. 

Water .... 

62.4 

0 

Mud. 

102. 

0-? 

Shingle, gravel . 

90-100-120 

35°-48° 

Clay .... 

120 

14°-45° 

Gravel and earth, 

126 

- 

Settled earth . . 

120-137 

21°-37° 

Dry sand . . . 

90 

34° 

Damp sand . . 

120-128 

35°-45° 

Marl. 

100 

- 

Brick .... 

90-135 

- 

Mortar .... 

86-110 

- 

Brickwork . . 

110 

- 

Masonry . . . 

110-144 

- 

Sandstone . . . 

130-157 

- 

Granite . . . 

164-172 

— 


We may assume generally, as safe values 
for brickwork, 110 pounds per cubic foot; 










6 


and for walls, one-half ashlar and one-half 
rubble backing, of granite 142 pounds, and 
of sandstone 120 pounds per cubic foot, 
though the last two values are generally 
exceeded. For ordinary earth or sand filling 
the angle of repose can be taken at one and 
one-half base to one rise, or a slope of 
33°42' with weights per cubic foot varying 
from 100 to 130. 

It is always advisable, where practicable, 
to put a layer of shingle next the wall, and 
to consolidate the layers of the filling by 
punning or other means, so as to reduce the 
natural slope as much as possible. 

With a well-built wall, designed after 
methods to be given ; having a good foun¬ 
dation-course, larger than the body of the 
wall, to better distribute the pressure, and 
resist sliding, and backed as described ; with 
weeping holes near the bottom at intervals, 
— there should be no fear of failure under 
ordinary conditions. 

G. It would take us too far to enter into 
the history of the theory of the retaining- 
wall. On this point see an interesting article 
by Professor A. J. DuBois in the “Journal 


of the Franklin Institute’’ for December, 
1879, on “ A New Theory of the Retaining- 
Wall.” In this work three methods will be 
developed and tested by the experiments 
recorded. Two of these methods are 
founded on the recent extended experi¬ 
ments of Leygue (“Annales des Fonts et 
Chaussees ” for November, 1885) and 
others, and the third is deduced by aid of 
the mechanical laws of stability in a granu¬ 
lar mass. 

7. In case a wall moves forward, how¬ 
ever little, or there is settling of the earth 
behind it, the earth generally rubs against 
the back of the wall, thus developing fric¬ 
tion. There are, however, certain inclina¬ 
tions of the back of the wall that will be 
specially examined in articles 28-31, for 
which the earth sooner breaks along some 
interior plane, in its mass, than along the 
wall, so that a certain wedge of earth will 
move with the wall as it overturns or 
tends to move. For all other cases, which 
include nearly all the cases in practice, 
there will be rubbing of the earth against 
the wall, so that the earth-thrust against 


the wall must be assumed to make, with 
the normal to the wall, an angle equal to the 
co-efficient of friction of earth on wall, 
unless this is greater than for earth on earth, 
in which case any slight motion of the wall 
forward will carry with it a thin layer of 
earth, so that the rubbing surfaces are those 
of earth on earth. 

8. These suppositions are found to agree 
with experiments. The old theory that 
assumed the earth-thrust as normal to the 
back of the wall, or, as in Rankine’s theory, 
always parallel to the top slope, does not so 
agree, aqd, in fact, often gives, for walls 
at the limit of stability, the computed thrust 
as double that actually experienced. The 
true theory, therefore, includes all the fric- 
tiou at the back of the wall that is capable 
of being exerted. This friction, combined 
with the normal component of the thrust, 
gives the resultant earth-thrust inclined 
below the normal to the back of the wall at 
the angle of friction to this normal. 1 


1 In Annales des Ponts et Chaussees for April, 1887, 
M. Siegler has given the results of 6oiue simple experiments 
proving the existence of a vertical component of the earth- 



9 


9. Rankine’s assumption that the direc¬ 
tion of the earth-thrust is always parallel to 
the top slope applies only to the case of an 
imaginary incompressible earth, homogene¬ 
ous, made up of little grains, possessing 
the resistance to sliding over each other 
called friction, but without cohesion ; of in¬ 
definite extent, the top surface being plane ; 
the earth resting on an incompressible foun¬ 
dation, or one uniformly compressible, and 


thrust against the movable side of a box filled with sand, by 
actually measuring the increased friction at the bottom of 
the movable board, held in place, caused by this vertical com¬ 
ponent. # The box was one foot square at the base; and for 
successive heights of sand of one-third, two-thirds, and one 
foot, the vertical components of the thrust for earth level at 
top was 0.66 pound, 1.76 pounds, and 3.97 pounds, respec¬ 
tively. Similarly for a box, 0.5 x 0.8 feet, filled with sand, 
but having a movable bottom supported firmly on iron blocks, 
the force necessary to move the blocks under the sides and 
under the bottom was measured; and from this the relative 
weights of sand supported by the bottom and sides of the 
box was found to be as one to one, nearly, for a height of 
sand of 0.6 foot, and about two to one fora height of 1.18 
foot, the total weights ascertained by the friction apparatus 
also checking out with the actual to within five per cent. 
Other experimenters have actually weighed the amounts held 
up by the sides and bottom, respectively. See Engineering 
News for May 15 and 29, 1886, also the issue for March 3, 
1883, ou “A Study of the Movement of Sand;” also see 
article 60 following. 



10 


being subjected to no external force but its 
own weight. 

For such a material, the only pressure 
which any portion of a plane parallel to the 
top slope of greatest declivity can have to 
sustain is the weight of material directly 
above it; so that the pressure on the plane 
is everywhere uniform and vertical. If we 
now suppose a parallelopipedical particle, 
whose upper and lower surfaces are planes 
parallel to the top slope, and bounded on 
the other four sides by vertical planes, we 
see that the pressures on the upper and 
lower surfaces are vertical, and their differ- 

0 

ence is equal, opposite to, and balanced by 
the weight of the particle. It follows that 
the pressures on the opposite vertical faces 
of the particle must balance each other 
independently, which can only happen when 
they act parallel to the top surface, in which 
case onty are they directly opposed. The 
pressures, therefore, on the two vertical 
faces parallel to the line-of greatest declivity 
will be horizontal; and on the other two 
faces, parallel to the line of greatest de¬ 
clivity. This is Rankine’s reasoning, and 


11 


it is sound for the material and conditions 
assumed. It is likewise applicable to a 
material of the same kind, only compressible, 
provided we suppose it deposited, as snow 
falls, everywhere to the same depth, on an 
absolutely incompressible, or a uniformly 
compressible, plane foundation, parallel to 
the ultimate top slope of the earth ; for then 
the compression is uniform throughout the 
mass, and does not affect the reasoning. 
But if we suppose, as usually happens, that 
the foundation is not uniform in compressi-' 
bility, then the earth will tend to sink where 
it is most yielding. This sinking is resisted 
to a certain extent by the friction resulting 
from the thrust of the earth surrounding the 
falling mass, so that much of its weight is 
transmitted to the sides, as actually happens 
in the case of fresh earth deposited over 
drains, culverts, or tunnel linings which 
settle appreciably. In the case of a tunnel 
driven through old ground, most if not all 
the weight of the mass above it is trans¬ 
mitted to the sides ; at least, at first, before 
the timbering or masonry is got in. Again, 
if the mass of earth is of variable depth. 


12 




even on a firm foundation, the mass of 
greatest depth will sink most, thus trans¬ 
mitting some of its weight to the sides, so 
that throughout the entire mass the press¬ 
ure is nowhere the same at the same depth 
as assumed. The vertical pressure over a 
drain or small culvert crossing an ordinary 
road embankment is less, too, for another 
reason, where the embankment is highest. 
The earth-thrust on a vertical plane, parallel 
to the line of road, is horizontal for a sym¬ 
metrical section when the plane bisects that 
section. On combining this thrust with the 
weight of the material on either side, we 
see that the resultant load on the culvert is 
removed farther from the centre than if 
there was no horizontal thrust. It is on 
account of this tendency to equalize press¬ 
ure by aid of the friction resulting from the 
earth-thrust, that sand, when it can be con¬ 
fined, is one of the best foundations, whether 
in mass or in the form of sand piles. 

10. In the case of earth deposited behind 
a retaining-wall on a good foundation, the 
settling of the earth will generally be greater 
than that of the wall, so that the earth rubs 


13 


against the wall, giving generally the direc¬ 
tion of the thrust no longer inclined, even 
approximately parallel to the top slope 
(except when the latter is at the angle of 
repose), but making with the normal to the 
back of the wall an angle downwards equal 
to the angle of friction. If the wall should 
settle more than the filling, the thrust would 
at first have a tendency to be raised above 
the normal. But if such a thrust, when 
combined with the weight of the wall, passes 
outside of the centre of the base of the 
wall, the top of the wall will move over 
slightly, the earth will get a grip on the wall 
in the other direction ; so that it is plainly 
impossible for the wall (for usual batters at 
least) to overturn or slide on its base, with¬ 
out this full friction, acting downwards at 
the back of the wall, being exerted. Hence 
the theory which supposes it is safe ; for 
although it is possible that the earth may 
make the effort at times to exert the full 
thrust given by Rankine’s formula, yet this 
effort is suppressed instanter by the external 
force now introduced by the wall friction, 
which force was expressly excluded from 


14 


the Rankine tlieory. The exceptions to 
this rule will be noted in article 31. 

11. Weyranch’s objections to taking the 
thrust inclined at the angle 4>' of friction 
to the normal are easily met. He says. 
Take a tunnel-arch ; and if we suppose the 
pressure, as we go up from either side, to 
make always the angle <f>' with the normal, 
we shall have at the crown two differently 
directed pressures : similarly for a horizon¬ 
tal wall with level-topped earth resting on 
it. If there is no relative motion, or ten¬ 
dency to motion, the thrust in the latter 
case is of course vertical, and in the former 
is probably vertical at the crown and in¬ 
clined elsewhere ; but if the arch or wall 
moves, and there is rubbing of the earth 
on the masonry, there is necessarily friction 
exerted ; so that the thrust at any point can 
have but one direction, making the angle <f> 
with the normal. 

12. Mr. Benjamin Baker, in his paper 
before the Institution of Civil Engineers, 
on the “Actual Lateral Pressure of Earth¬ 
work ” (republished by Van Nostrand as 
“Science Series,” No. 5G), tested an old 


15 


theory (where the earth-thrust was assumed 
to act normal to the wall) by the results 
of experiments, and found the theoretical 
pressure often double the actual. In the 
discussion which followed, not a single 
engineer so much as alluded to a truer 
theory which assumes the true direction of 
the earth-thrust, and has been known and 
used, just across the channel, since the 
time of Poncelet. 

The writer tested this theory by many of 
the experiments recorded by Baker and 
some others, and found it to agree, within 
certain limits, remarkably well (see “Van 
Nostrand’s Magazine ” for February, 1882). 
These results have been carefully revised, 
and new experiments included, in the table 
given farther on, from which the reader 
can form a fair estimate of the theory as a 
working theory within certain limits that 
will be indicated. 

The reader is referred, however, to Mr. 
Baker’s essay, not only for experiences 
under ordinary conditions, but for those 
exceptional cases which seem to defy all 
mathematical analysis. In fact, the engi- 


1G 


neer almost invariably has to assume the 
weights of earth and masonry, and angle 
of repose of the earth. Where there is 
water, the conditions one day may be very 
different from what they are the next, 
especially if the foundation is bad, as often 
happens ; in which case the wall will move 
over simply on account of the compres¬ 
sibility of the foundation, so that it has 
perhaps nothing like the estimated stability. 
For all such cases an allowance must be 
made over the results given for a firm 
foundation, etc., as to which no rule can 
be given. 

O t 

As water often saturates the filling, and 
perhaps gets under the wall, we must con¬ 
sider, in certain cases, water-pressure in 
connection with the thrust of the backing. 
Therefore, a short chapter on reservoir- 
walls, or dams, follows, in which many of 
the principles that must likewise apply to 
retaining-walls proper are given. 


17 


CHAPTER I. 


RESERVOIR-WALLS. — 

u $ rcCUMETO WWW 

*13. The design of reservoir-walls is a 
subject that has received the attention of 
many engineers and mathematicians; but 
they are by no means agreed, except in a 
general way, upon the precise profile thp,t 
is best to satisfy, as uniformly as possible, 
the requirements of strength and stability. 

We shall very briefly, and by the shortest 
means, point out the main principles of 
design of a dam that resists overturning 
or sliding by its weight alone, and is called 
a gravity dam, in contradistinction to one 
built on a curve that requires the aid of 
arch action to render it stable. 

Let Fig. 1 represent a slice of the dam 
contained between two vertical parallel 
planes one foot apart, and perpendicular to 
the faces. 

When the dam is large, a roadway is 


generally built on top, so that the faces ks 
and gi are vertical or nearly so for some 
distance down ; after which the profile is 
designed to meet certain requirements, to be 
given presently. Let us suppose that the 


Figi-l 



PoCTTrn 
J>i+P 2 


1 jij 


:~z=±± 

VW 




dam has been properly designed down to 
the horizontal joint df and that the weight 
of the portion above df equals W v regard¬ 
ing the weight of a cubic foot of masonry 
as 1, and that its resultant cuts the joint 
df at the point o. 

































19 


To design the part fabd below df by a 
rapid though tentative method, we must 
first assume the slopes db and fa corre¬ 
sponding to the depth do; then compute the 
areas of the triangles bed and afe, and of 
the rectangle feed. The distances of the 
centres of gravity of these areas (which 
represent volumes) from the point b are re¬ 
spectively | be, be -f- ^ae, and be -f- \ee. On 
multiplying each area by its correspond¬ 
ing arm from b, adding the product 
w, X (be -j- do), and dividing by the sura 
of W x (which equals the area of gkdf) 
and the portion added fabd, we find the 
horizontal distance bm from b to where the 
resultant of the weight above joint ab cuts, 
this joint. Its amount W is equal to the 
sum of the areas (IT, 4- abdf), and we 
have only to combine W acting along the 
vertical through m, with the horizontal 
thrust H of the water acting on the face 
Jcsdb, to find the resultant R on the joint* 
and the point n where it cuts that joint. 

There is a vertical pressure of the water 
on the part sdb; but, as it adds to the 
stability, it is generally neglected, particu- 


20 


kirly as the inner face is generally nearly 
vertical. 

14. The horizontal pressure of the water 
H for the height ft, by known laws of 
mechanics, is equal to the area ft X 1 mul¬ 
tiplied by the depth of its centre of gravity 

- below the surface of the water, and by 


the weight of a cubic foot of water w, 
where a cubic foot of masonry is taken as 
the unit. This pressure acts horizontally 
at ^ft above the joint aft, so that its moment 
about the point n where the resultant 1{ 

_ ft ft _ h s w 

cuts the 'base aft is ft . ^ * w * g ~ —j7“• 


The 


moment of W about the same point is 
W X mn. As these two moments must be 
equal, we find the distance between the 
resultant pressures on joint aft for reservoir 
empty and reservoir full, 


mn 


h*vo 

GM' 


The above is substantially one of the methods 
adopted by Consulting Engineer A. Fteley in the 
design of the proposed Quaker Bridge Dam. See 
his interesting report, and that of B. S. Cliurch, 





21 


chief engineer, with many diagrams of existing 
dams of large proportions, in “ Engineering News ” 
for 1888, Jan. 7, 14, Feb. 4, 11; also the discus¬ 
sions by the editor in the numbers for Feb. 4 and 
25, and March 3. 

15. There are three well-known condi¬ 
tions, that must hold at any joint if the 
profiles fa and db have been designed 
correctly: — 

1st, The points m and n where the re¬ 
sultants for reservoir empty or full cut the 
base ab must lie within the middle third of 
the joint or base ab. 

2d, The unit pressures of the masonry at 
the points a or b must not exceed a certain 
safe limit. 

3d, No sliding must occur at any point. 

1G. The last condition is evident, and 
requires that II < Wf where / is the co¬ 
efficient of friction of masonry on masonry, 
the adhesion of the mortar being neglected. 
If (f, is the angle of repose of masonry on 
masonry, / = tan <£, and we must always 
have, 


— < tan <±>; 
W 


that is, the resultant R must never make 
with the normal to the joint an angle 
greater than the angle of friction. In fact, 
in practice, we should employ some factor 
of safety as 2 or 3, so that 2// or 3 II should 
always be less than Wf. This third con¬ 
dition is of supreme importance at the 
foundation joints of dock-walls, which fail 
(when they fail at all) by sliding from 
the insufficient friction afforded by the wet 
foundation. For ordinary retaining-walls, 
too, the foundation should, when practi¬ 
cable, be inclined, so that R shall make a 
small angle with the normal to the base. 
In all cases, deep foundations are to be 
preferred, as the earth in front of the wall 
resists the tendency to slide appreciably. 

17. We shall now proceed to give a 
reason for the first condition above, and 
likewise deduce a formula to ascertain the 
unit stresses at the points a and b. 

If we decompose the resultant R at the 
point n , distant u = an from a (Fig. 1), 
into its two components II and IF, the 
former is resisted by the friction of the 
joint, and will be neglected in computing 


the stresses at a and b , though it doubtless 
affects them in some unknown manner. 
The remaining force W, acting vertically 
at n, must necessarily cause greater press¬ 
ure at the nearest edge than elsewhere on 
the joint, at least when the angle at a is not 
too acute, and the dam is a monolithic 
structure. For large dams built of stones 
in cement, it is likely that there will be 
greater pressure at the middle of the base 
than in a monolithic structure where the 
resistance to shearing or sliding along ver¬ 
tical planes is much greater than in a wall 
made up of many blocks, particularly if 
they are laid dry. But it is probably best, 
until experiment can speak more decisively 
on the point, to assume the pressure great¬ 
est at the toe nearest the resultant, and as 
given by the following theory : — 

Call l = length of joint ab 

u = an = distance from R to near¬ 
est toe ; 

then if we suppose applied at the centre of 
the joint two vertical opposed forces, each 
equal to W , it does not affect equilibrium. 
We- can now suppose the force W acting 


2 4 


downwards at the centre to be the resultant 

W 

of a uniformly distributed stress p 1 =—, 

shown by the little arrows just below joint 
db; and that the remaining forces IF, one 
at the centre and one at n , acting in oppo¬ 
site directions, and constituting a couple* 
whose moment is W l — u ), cause a uni¬ 
formly increasing stress, as in ordinary 
flexure (shown by the little arrows below 
the first), wdiose intensity at a or b is by 
known laws, 


Vi= M = Wl 



= w 


3 1 — 6u 

~T 2 


The total stress p at the nearest toe a is 
therefore the sum of p x and jo 2 , and is com¬ 
pressive. 


•. p = 



3 n\W 
i )i ■ 


■ ■ (i) 


The stress at b is of course p x — p 2 , where 
this is not minus indicating tension, unless 
the joint can stand the tension required. 
If we call u' the distance from n to the 
farthest toe, i.e. u' = nb , we have the mo- 





ment of the two weights W = W(v! — %l ). 
On substituting this value for — u) in 
the value for p 2 above, we find for the unit 
stress at b the identical equation (1) above, 
provided we replace u by u '; so that the 
equation is general, and applies to either 
toe, if we only substitute for u the distance 
of the resultant from that toe. The stress 
is distributed, as shown by the lower set of 
arrows in Fig. 1, where there is only com¬ 
pression on the joint as should always 
obtain. The stress is thus uniformly in¬ 
creasing from the right to the left. If the 
limit of elasticity is nowhere exceeded, it 
follows that a plane joint before strain, will 
remain a plane joint after strain, as must 
undoubtedly be the rule for single rectangu¬ 
lar blocks. 

Referring to equation (1), w T e see that if 
we replace u by v! = 1 1, that the stress at b 
is zero, from which point it increases uni¬ 
formly to a, where its intensity, for u = 



or twice the mean. 


For greater 


values of u' than f l, the stress at b becomes 
tensile, which is not desirable; hence the 


20 


reason for condition 1 above, that the re¬ 
sultant should lie within the middle third 
of the joint. 

If the joint cannot resist tension at all, 
and R strikes outside the middle third, the 
joint will bear compression only over a 
length 3m, and the maximum intensity at 
W 

a is now 2—. This is evident, if we treat 

3 m 

3m = V as the length of joint, and substi¬ 
tute this value for l in formula (1). There 
is now no pressure at the distance 3m = V 
from the left toe by the previous reasoning 
for the original joint l , and to the right of 
that point the joint will open, or tend to 
open. It is evident for full security that 
the resultant should strike within the mid¬ 
dle third some distance to allow for con¬ 
tingencies. 

18. Having computed the unit pressures 
at the nearest toes for reservoir full or 
empty, condition 2 requires that these 
pressures do not exceed certain limits : in 
case they do, the lower profiles have to be re¬ 
vised, and the computation above repeated* 
until all the conditions are satisfied. 


\ 


In the proposed design for Quaker Bridge 
Dam, maximum pressures per square foot at the 
toes, at the base, were limited to 30,828 lbs. at 
the backhand 33,266 lbs. at the face; these 
pressures diminishing gradually to one-half to 
within about 100 feet from the top, the total 
height of dam from the foundation being 265 
feet; the argument being that the lower parts 
could stand more pressure than the upper parts 
shortly after construction, on account of the 
cement there attaining a greater strength. Be¬ 
sides, for this unprecedented height of dam, to keep 
the lower pressures within more usual limits “it 
would be necessary to spread the ■ lower parts in 
an impracticable manner, and to incline the slopes 
to an extent incompatible with strength.” 

It is evident that by this method of design 
there is no fixed rule by which any two computers 
could arrive at the same profile, having given the 
upper part empirically, sufficient in section to 
carry a roadway, and to resist the additional 
stresses due to the shock of waves and ice, at a 
time, too, when the mortar is not fully set. 

Such a rule is most easily introduced by 
requiring a certain factor of safety against over¬ 
turning, and, moreover, that the factor of safety 
against sliding along any plane shall not fall below 
a certain amount. It is suggested, however, that 
the factors of safety should increase from the 
foundation upwards, to make the section equally 
strong everywhere against overturning, when 


28 


allowance is made for the effects of wind and wave 
action, floating bodies, the expansive force of 
ice, or perhaps the malicious use of dynamite. 
If this is admitted, it would add one more con¬ 
dition (4) to the three previously stated, and 
would secure greater uniformity in design. See 
Appendix. 

As to the unit pressure test (condition 2), it 
must be observed, that we know little or nothing 
as to what limit to impose; for not only is the 
stress all dead load (which would allow of higher 
unit stresses), but the unit resistance of masonry 
in great bulk is undoubtedly much greater than 
in small masses (not to speak of tests on small 
specimens as a criterion), since the shearing off 
which follows, or is an incident to, crushing can 
hardly occur in the interior of a large mass of 
masonry. 

ID. AVe shall find in the end, that, for 
different forms of retaining-walls to sustain 
earth, that a factor of safety of about 2.5 
against overturning is highly desirable, and 
that it will generally satisfy the middle third 
limit. In such walls this factor must be 
introduced to provide against an actual 
increase of the earth-thrust, due to water, 
freezing, accidental loads, and above all 
to the tremors caused by passing trains or 


29 


vehicles (if these are not considered sepa¬ 
rately), which it is well-known have caused, 
by increased weight, and the increased 
pressure due to lowering the natural slope, 
a gradual leaning and destruction of walls 
of considerable stability for usual loads. 

In a very high dam this is different: 
the pressure rarely changes but little, ex¬ 
cept on the upper portions ; so that, if such 
conditions were to hold indefinitely, the 
limiting unit stresses should control the 
lower profile more than a factor of safety 
against overturning. But, as pointed out 
by the editor of “ Engineering News ” (in 
the issues above referred to), a dam on 
which the fate of a city may ultimately 
depend should be designed, as far as pos¬ 
sible, to resist earthquakes also. For that 
contingency, there is a reason for the factor 
of safety against overturning and sliding 
being as great as possible throughout; and 
by putting the gravity dam in the arch 
form, convex up stream, the resistance to 
earthquake and other shocks is enormously 
increased. 

20. We have now given the general prin- 


30 


ciples that should guide in the design of 
dams, which likewise apply in the design 
of retaining-walls proper, where, however, 
the height is rarely sufficient to call for 
much, if any, change of profile, and the 
maximum pressures are usually far within 
safe limits when a proper factor of safety 
against overturning or sliding has been 
introduced, which satisfies likewise the con 
dition that the resultant shall cut the base 
within the middle third. We of course 
have, as stated before, the direction of the 
earth-thrust inclined below the normal to 
the wall at the angle of friction ; otherwise, 
the methods above are applicable when the 
value of that earth-thrust has been deter¬ 
mined. For dock or river walls, saturated 
with water, we must combine the water- 
thrust with the earth-thrust. 

If we suppose the filling of gravel, the 
water surrounding each stone allows free¬ 
dom of motion ; but the weight of the solid 
stones of the filling must now be taken less 
than when in air, by the weight of an equal 
volume of water, or at the rate of 62.4 
lbs. per cubic foot (or say 64 for salt 


water), and the earth-thrust then found 
for the angle of repose of stone lubricated 
with water. Thus, if the weight of the 
solid stone be 150.4 lbs. per cubic foot, 
and the voids are thirty per cent, the weight 
of solid stone in water is 88 lbs. per cubic 
foot, and that of the tilling 88 x .70 = 
61.6 lbs. in water, although it was 105 in 
air. 

If the wall is founded on a porous 
stratum, the weight of the masonry is sim¬ 
ilarly reduced by 62.4 lbs. per cubic foot, 
or say one-half ordinarily; but if the 
foundation is rock or good clay, “there is 
no more reason why the water should get 
under the wall than it should creep through 
any stratum of a well-constructed masonry 
or puddle-dam,” as Mr. Baker has ob¬ 
served. 

If the water cannot get in behind the 
wall, the water in front only assists the 
stability. 

It has been previously observed that 
sliding is principally to be guarded against 
in dock-walls and others similarly situated, 
which can only be done by a sufficient 


weight of masonry irrespective of its shape, 
unless the foundation is inclined, which 
even in the case of piling has been effected 

Fig. 2 



by driving the piles obliquely, of course 
as nearly at right angles to the resultant 
pressure as is practicable. 

Fig. 2 represents a wall with a curved 
batter, in brickwork with radiating courses, 

O 7 












that might be used for a quay or river-wall, 
or a sea-wall, as ships can come closer to 
the brink than in the case of a straight 
batter; besides, for sea-walls it resists the 
action of the waves better. The centre of 
gravity can be found by dividing the cross 
section up into approximate rectilinear 
figures, and proceeding as in finding the 
position of W in Fig. 1. Its position is 
a little farther back than for a straight 
batter, which adds to its stability. But it 
is difficult to construct, the joints at the 
back are often thicker than is advisable, 
and there is probably no ultimate economy 
in its use. 


34 


CHAPTER II. 

THEORY OF RETAINING-WALLS. 

Graphical Method. 

21. In the theory of earth-pressure that 
follows, we shall consider the earth as a 
homogeneous, compressible mass, made up 
of particles possessing the resistance to 
sliding over each other called friction, but 
without cohesion. This is a much simpler 
definition than the one that Rankine’s 
theory calls for (see Art. 9), and is more 
true to nature; the only approximation, in 
fact, consisting in neglecting cohesion, if 
we consider a homogeneous earth like dry 
sand. 

Let Fig. 3 represent a vertical section of 
a retaining-wall ABCD, backed by earth, 
whose length perpendicular to the plane of 
the paper is unity. 




35 


Assumption. We assume that the earth 
behind the wall, whether the top surface is 
a plane or not, has a tendency to slide 
along some plane surface of rupture as 
Al, A'2, ... . 


Tri g.. 3 



No proof is given of this assumption, so 
that it can only be tested by experiment; 
but for the present we shall adopt it. 

In connection with the hypothesis of a 
plane surface of rupture, we shall use only 
one principle of mechanics relative to the 












stability of a granular mass, first stated 
by Rankine as follows : — 

It is necessary to the stability of a 
granular mass , that the direction of the 
pressure between the portions into ivhich it 
is divided by any plane should not , at any 
point , make ivith the normal to that plane 
an angle exceeding the angle of repose. 

This principle will alone enable us to 
ascertain the earth-thrust against any plane 
without resorting to a special principle, 
like Coulomb’s “ wedge of maximum 
thrust,” which last, however, will be in¬ 
cidentally demonstrated as a consequence 
of the above law. 

22. In Fig. 3, let us consider the 
triangular prisms CC10, CA\, . . . , as 
regards sliding down * their bases .40, 
Al, . . . 

If AF is the natural slope of the earth, 
the tendency of the prism CAF to slide 
along AF is exactly balanced by friction, 
as is well known. But if we consider 
other possible planes of rupture, lying 
above AF. as ylO, yll, . . . , we see, unless 
the wall offers a resistance, that sliding 


37 


along some one of these planes must 
occur: so that the earth exerts an active 
thrust against the wall, which must be 
resisted by it; otherwise, overturning or 
sliding would ensue. 

In case the wall is subjected to a thrust 
from left to right, as from earth, water, 
etc., acting on BD , and this thrust is 
sufficient to more than counterbalance the 
active thrust of the earth to the right of 
the wall, it will bring in the passive 
resistance of the earth to sliding up some 
plane as A'l , and the surface of rupture 
will now resist motion upwards, in place of 
downwards as hitherto. 

In the first case, of active thrust, where 
the prism is just on the point of moving 
down the plane, we know by mechanics 
that the resultant pressure on the plane 
of rupture makes an angle </> of f riction of 
earth on earth with the normal to that 
plane and directed below the normal; in 
the second case, of passive thrust, the 
direction of the pressure lies above or 
nearer the horizontal than the normal, and 
makes the angle </> with the latter. 


38 


23. Td the first case, where the wall 
receives only the active thrust of the prism 
of maximum thrust, let us call G (Fig. 3) 
the weight in pounds of this prism ; S the 
resultant pressure on the surface of rupture, 
making an angle </> with the normal to that 
plane below the normal; and E the resultant 
earth-pressure on the wall, which (except 
for cases to be noted in Art. 31) makes an « 
angle c ft of friction of earth on wall with 
the normal to the wall below the normal, 
unless (fV > cf), in which case a thin layer 
of earth will go with the wall, in case of 
relative motion, and this layer rubbing 
against the remaining earth will only cause 
the friction of earth on earth, and E will 
only be directed at an angle (f> below the 
normal; supposing always that the tendency 
to relative motion corresponds to the earth 
moving down along the back of the wall 
AC, as in settling from its compressibility, 
or as in case of an incipient rotation of the 
wall forward, from a greater pressure on 
the outer toe or a slight unequal compres¬ 
sion of the foundation. 

It remains to find the position of the true 


plane of rapture. As preliminary to this, 
we note from Fig. 3 an expeditious way of 
finding the direction of S on any trial plane 
of rupture, as Al. Thus calling w the angle 
that Al makes with the vertical A/, the 
straight line making an angle (<£ -f w) 
with any horizontal, as DCI, below that 
horizontal, is parallel to S, since any line 
inclined at an angle m below the horizontal 
is perpendicular to Al, and S is inclined at 
an angle <f> below that normal. In laying 
off the equal angles, it is convenient to use 
a common radius, AH, to describe the arcs 
having A and / respectively as centres, and 
to take chord distances of the arcs </> and w, 
and lay them off on the arc with I as a 
oentre, as shown. For any other trial 
plane, as A2, we have simply to lay off the 
corresponding value of w below the angle cf> 
as before. 

24. We shall now refer to Fig. 4, to 
illustrate the general method to follow to 
find the earth-thrust E in pounds. Here 
the wall, one foot long perpendicular to the 
plane of the paper, is shown in section 
BACD , the earth sloping at an angle from 



40 



some point on the top of the wall to the 
point marked 2, where it is horizontal. 
This is called a surcharged wall, the earth 











41 


lying above the horizontal plane of the top 
of the wall being called the surcharge. 

Extend the line AC of the inner face to 
0, where it intersects the top slope of the 
earth ; the possible prisms of rupture are 
then .401, 402, 403, . . . , and we shall 
now proceed to reduce these areas to equiv¬ 
alent triangles having the same base 42. 
Draw the parallels 00', 11', 33', . . . , to 
line 42 to intersection with a perpendicular 
to 42, passing through the point 2. Then 
the triangle 402 is equivalent to the triangle 
A0'2, and 412 to 41'2, so that triangle 
A0'l' is equivalent to 401. Similarly 426 
is equivalent to triangle 426'4, having the 
-same base, -4.2, and vertices in a line parallel 
to this base, giving the same altitude. Thus 
the area 40264 is replaced by 40'6'4 ; and 
the weight of the corresponding prism, if we 
call e the weight per cubic foot of earth, 
is \AA x 0'6' X e. Similarly the weight of 
4024 is %A2 x 0'4' x e ; so that if we use 
O'T, 0'2, 0'3', . . . , to represent the weights • 
of the successive prisms -4.01, 402, 403, 

. . . , on the force diagram given below, 
we have simply to multiply the value of E, 


42 


given by construction, by £e,A2 to find its 
true value in pounds. 

We next lay off the successive values of 
(c ft -f- w), as in Fig. 3. Thus, with any 
convenient radius, as ^10, we describe an 
arc, ogdf ,, and call the intersections with 
A 1, A’2, . . . , a v a 2 , « 3 , . . . , respectively. 
Next, through point g on the arc in the 
vertical through A, draw vertical and hori¬ 
zontal lines, and describe an arc, bss v . . . 
with the same radius ; then draw gs , making 
the angle below the horizontal gb (by 
making chord bs = chord fd) , and lay off 
with dividers, chords ss v ss 0 , ss 3 , . . . , 
equal to chords ga v ga 2 , ga v ... It is 
evident now that lines f/.sq, t </.s* 2 , gs s , . . . , 
make the angles </> with the normals to the 
successive planes A\ , A’2, A3, . . . . and thus 
give the direction of the S ’s corresponding 
to those planes. 

We now lay off with dividers on the 
vertical line gA the distances gg v gg . 2 , . . . 
equal respectively to OT', 0'2, 0'3', . . . r 
and draw through the points g v g 2 , g 3 , 
parallels to the direction of E (drawn as 
before explained) to intersection with the 




lines t/Sp gs 2 , gs a , . . . , which intersections 
call c r c 2 , c 3 , . . . , respectively. 

25. It follows that the lines g x c v g 2 c 2 , 

J7 8 c 8 , . . . , represent the thrusts E due to 

the successive prisms of rupture A01, ^402, 

. . . , and we shall now prove that the 

greatest of these lines, which is found to 

be g 4 c v represents the actual active thrust 

upon any stable wall. 1 This follows from 

the simple .fact, that if we regard any 

other thrust than the maximum as the true 

one, on combining this lesser thrust, taken 

as acting to the right, with the weight of 

the wedge of rupture corresponding to the 

maximum thrust, we necessarily find that 

the resultant falls below the position first 

assumed ; so that it makes an angle with the 

normal to the corresponding plane of rupture 

greater than the angle of repose, which, by 

the principle of Art. 21, is inconsistent with 

stability. Thus, in Fig. 4, if we choose 
* _ / 

to assert that any trial thrust, as g. 2 c 2 , less 
than the maximum g 4 c 41 is the true one, on 

1 This method of laying off the trial thrusts, so that the 
maximum could readily be obtained, was first given by 
Professor Eddy, in New Constructions in Graphical Statics. 







44 


shortening the lengths g B c B , g A c v . . . , 
representing superior thrusts, to the com¬ 
mon length g 2 c . 2 , and drawing through the 
new positions of c 3 , c 4 , . . . , straight lines 
to g, which thus represent the resultant 
thrusts on the planes A3, A4, . . . , we see 
that the new directions fall below the first 
assumed positions, and therefore make 
angles with the normals to the planes 
greater than 0, which is absolutely incon¬ 
sistent with equilibrium. It follows that 
any thrust less than the maximum, as 
determined by the construction above, is 
impossible; and that this maximum thrust 
thus found is the actual active thrust exerted 
against the wall. In this consists what is 
known as Coulomb’s “wedge of maximum 
thrust,” which is here established by aid of 
the single mechanical principle enunciated 
in Art. 21. 

The prism of rupture in this case is 
A024A, the plane A4 being the surface of 
rupture. 

To find the resultant thrusts on all the 
other assumed planes, we combine the actual 
thrust found with the weight of earth lying 



above the plane. Thus, extending g x c v g. 2 c 2r 
. . . to a common length g 4 c v or to the 
vertical tangent to the dotted curve, the lines 
drawn from g through the corresponding 
intersections with this vertical will represent 
the thrusts on the planes Al, A2, . . . , 
which are thus inclined nearer the horizontal 
than the old trial values, and thus make less 
angles than with the normals to their 
corresponding planes ; so that the conditions 
of stability are all satisfied, and, if the wall 
gives, sliding will only occur down the plane 
of rupture Al. 

In the analytical method followed by 
Weyraucli, E is assumed to be, constant, 
and to equal the actual thrust on the wall; 
and the real surface of rupture is taken to 
be that plane for which the angle that S 
(Fig. 3) makes with the normal is the 
greatest (<£) consistent with equilibrium, 
which is in agreement with what we have 
just proved 

Winkler adopts the same method, in 
preference to the Coulomb method. In 
fact, he asserts that “no author, from 
Coulomb down, has given any direct satis- 


40 


factory proof of Coulomb’s principle.” It 
is hoped that the above demonstration 
will prove complete and satisfactory. The 
method evidently gives the least thrust, for 
the assumed direction of E , that will keep 
the mass from sliding down the surface of 
rupture. 

The earth can resist a much greater 
pressure from the wall side, since a con¬ 
tinuously increasing pressure from the left 
•causes all the resultants on planes Al, A’2 , 
.. . . to approach the normals, then to pass 
them, and finally to lie above them, with 
the sole condition that none of them must 
make angles greater than <£ with their 
corresponding normals (see Art. 34). 

2G. To find the thrust E in pounds , we 
multiply g 4 c 4 to scale by' ^A'l.e. Finally, 
if we know the position of E, we combine 
it with the weight of the wall in pounds, 
acting along the vertical through its centre 
of gravity, to get the resultant on the base. 
If the upper surface of the earth is level, 
or with a uniform slope from the point 0 
(Fig. 4), then the sections of the prisms of 
rupture for various heights of the wall, or 


47 


for any values of AO, are similar triangles, 
so that the thrusts E , which vary directly 
with the weight of the corresponding prisms, 
will also vary directly as the areas of these 
triangles, or as the squares of the homologous 
lines AO, or as the squares of the height of 
point 0 from the base AB. It follows, as 
in the case of water-pressure, that for these 
cases the resultant E of the earth-thrust 
acting along the face AO is found at a 
point -JA0 along AO from the base AB. 

For the surcharged wall it is possibly 
higher; in fact, Scheffler takes it in con¬ 
structing his tables, for all cases, at t 4 qA 0 
along AO. But experiment indicates either 
that the thrust is overestimated for sur¬ 
charged walls, or that it acts not higher 
than at one-third the height of 0 above the 
base ; so that it will prove safe in practice 
to take the latter limit if we use the 
theoretical thrust. As to the latter, it is 
evident that cohesion (which we have 
neglected) has a greater area to act upon 
along the surface of rupture for any kind 
of surcharged wall, than for earth either 
level or sloping down from the top of the 


48 


wall; so that wo should expect the thrust to 
be somewhat overestimated when we neglect 
cohesion altogether, since the resistance to 
sliding down any plane due to it is directly 
as the area of the surface of separation. 

27. In case the earth is level with the 
top of the wall, the construction of Fig. 4 
again applies, only the line OV now coincides 
with the horizontal through C\ and the 
reduction of areas to equivalent triangles 
is omitted, since now all the triangles have 
the same altitude, equal to the height of the 
wall. 

If, however, the earth slopes uniformly 
from the top of the wall, at a less angle 
than the angle of repose, we can assume 
any point as 2, on this slope, and effect the 
construction of Fig. 4 as before; or, better, 
we can divide this slope into a number of 
parts at 1, 2, ... , and treat 01, 02, . . . , 
successively as the bases and the perpen¬ 
dicular from A upon 02, produced as the 
common altitude; so that, using 01, 02, 
. . . , as representing the weights of the 
corresponding prisms on the load line gg , 
we have finally to multiply the value of gc, 



49 


corresponding to the greatest thrust, by 
Je, multiplied by this perpendicular, to get 
the maximum thrust E in pounds. 

In case the surface of the earth slopes 
indefinitely at the angle of repose , the 
graphical method fails to find the surface 
of rupture, which analysis shows, in this 
case, to approach indefinitely to the plane 
of natural slope passing through the point 
A , though practically it may be shown that 
planes of rupture slightly above the latter 
will give almost identically the same earth- 
thrust, so that they can safely be used. In 
fact, it is well to state here, that', for earth- 
level at top, the surface of rupture, as 
observed in experiments with every kind of 
backing, agrees very well with theory ; but, 
as the surcharge grows higher, the actual 
surface of rupture lies nearer the vertical 
than the theoretical, and the thrust is 
correspondingly less, particularly for walls 
leaning backwards at top, which, for a high 
surcharge, actually receive much less thrust 
than the simple theory after Coulomb’s 
hypothesis, neglecting cohesion, calls for; 
ind it is not surprising that it is so. But 


we shall defer the comparison of numerical 
results till later. 

28. Case where E does not make the angle 
<f> or <// with the normal to the wall. 

In Fig. 5, let AC represent the inner face 
of the wall, backed by earth sloping upwards 
from C in the direction (7—10. There 
are certain positions for the wall AC lying 
to the left of the vertical Ag , for which the 
true thrust on it is found by ascertaining 
the thrust on the vertical plane ylO, extending 
from the foot of the icall A to where it 
intersects the top slope C — 10, having 
assumed the direction of the thrust on AO , 
after Rankine , as parallel to the top slope , 
and combining this thrust , acting at -J^IO 
above A , with the iceight of the mass of 
earth , ^40<7, lying between AO and AC. 
acting along the vertical through its centre 
of gravity. The thrust on ylO is thus 
combined with the weight of AOC, at a 
point on AC, one-tliird of its length going 
from A to C. 

This direction of the thrust on AC par¬ 
allel to the top slope is in agreement with 
Rankine’s principle for the ease of an 


51 


unlimited mass of earth of the same depth 
everywhere, on an uniformly compressible 
foundation (Art. 9), and doubtless agrees 
very nearly with the direction and amount 
of the earth-thrust in ordinary eases, except 
near comparatively rigid retaining-walls, or 
other bodies, where the direction is generally 
changed, as previously pointed out. Let 
us ascertain the limiting position of AC, ' 
below which the true thrust must be ascer¬ 
tained in the manner just stated. To do 
this, we first assume the thrust on AO as 
acting parallel to the top slope, and find 
its intensity corresponding by previous 
methods ; and afterwards prove, for positions 
of AC below the limit, to be found b} T 
construction, that no thrust on AO having 
a less inclination to the vertical is consistent 
with equilibrium. 

The construction necessary to find the 
thrust on AO, from the earth on the right, 
is similar to that given for Fig. 4, except 
that the top slope is now uniform, and will 
only be briefly indicated. Thus, divide the 
top slope 0 — 10, to the right of Ag, into a 
number of parts, made equal for convenience, 


and draw through the points of division lines 
from A produced on to meet the arc described 
with Ag as a radius at the points a v a 2 , . . . 
Then with g as a centre, and gA as a radius, 
describe a semicircle as shown ; also draw 
gb horizontal, and lay off arc bs equal to <£, 
the angle of repose, and from s lay off arcs 
ss v ss 2 , . . . , equal to ga v ga 2 , and draw 
the lines gs v gs 2 , . . . , from g through 
the extremities of these arcs to represent the 
directions of the resultants on the successive 
planes of rupture, which are thus inclined 
below the normals to those planes at the 
angle (j> respectively. Next, on the vertical 
gA , lay off gg v gg. 2 , . . . , equal to the bases 
01, 02, of the supposed prisms of rupture 
lying to the right of Ag, and through their 
extremities draw g x Y, g., 2', . . . , parallel 
to top slope to intersection 1', 2 ', . . . , 
with the directions of the resultants first 
found. The greatest of these lines cc', to 
scale, represents the actual thrust on AO ; 
and we have only to multiply it by \ep, 
where p is the perpendicular let fall from 
A on the top slope 0—10 produced, to scale, 
to get the pressure in pounds, if desired. 



Now, if the direction of the pressure on the 
wall AC cannot be taken as usual, inclined 
below the normal to AC , at an angle </>, 
it is (Art. 7) because, in case of motion, 
the earth does not rub against the wall 
sufficiently to develop the required friction, 
whence it must follow that the earth breaks 
along some plane as A4, Ah , . . . , to the 
left of Ag, where the thrust is inclined at 
the angle <£ to its normal; so that this plane 
is a veritable plane of rupture , and its 
position can be found as usual on assuming 
the direction of the thrust on AO as parallel 
to the top slope. 

In case such, a plane exists between AO 
and AC , the earth below it, if the wall 
moves, will go with the wall; further, it is 
evident that the thrust against the vertical 
plane AO, due to the wedge of rupture on 
the left, must exactly equal the thrust first 
found corresponding to the wedge of rupture 
on the right, otherwise equilibrium will be 
impossible. 

To ascertain the position of this plane of 
rupture on the left, that we shall hereafter 
call the limiting plane , most accurately, it 


54 


is well to magnify the lines representing the 
forces as much as the limits of the drawing 
will admit of. We have consequently 
divided the top surface, 06’, into a number of 
equal parts, of which the first eight are only 
one-fourth the length of the corresponding 
parts to the right of AO. By laying off the 
loads gg v gg 2 , . . . , however, to a scale 
four times as large as just used, we have 
the lengths gg v gg 2 , . . . , exactly four 
times the lengths 01, 02, . . . , along the 
surface to the left of 0, so that the old 
lettering applies again. 

We now produce the lines Al, ^42, . . . , 
to intersection n v n 2 , . . . , with the arc gn 
(it is obvious that the top slope, 0(7, should 
best be drawn, in the first instance, through 
g , for accurately fixing the positions of n v n 2 , 
. . .) ; then lay off, below the horizontal, 
the angle dgm = ; and from m, the inter¬ 

section of gm with the semicircle clAb, lay 
off the arcs mm v mm 2 , . . . , equal to 
gn v gn 2 , . . . : so that the lines gm v gm 2 , 
. . . , all make angles equal to with the 
normals to the corresponding planes Al, 
A2, . . . 




Next, on drawing through g v g 2 , . . . , 
lines parallel to the assumed direction of 
the thrust on AO, to intersection with the 
corresponding lines gm v gm 2 , . . . , the 
greatest of the intercepts (f/ 5 f 5 nearly) 
represents, to the scale of loads, the thrust 
on the plane AO ; and this length should 
exactly equal four times the length cc 7 
representing the thrust from the right, as 
we find to be the case. The plane of 
rupture to the left of the vertical through 
A thus coincides nearly with Ad, which is 
marked “limit” on the drawing. [On a 
larger drawing, for (ft == 33° 42' and the top 
sloping at 25°, the limiting plane was found 
to make an angle of 15° to 16° (see a more 
accurate determination in Art. 41) to the 
left of the vertical Ag, and to lie slightly 
below Ad, as this drawing would indicate.] 
If we lay off along the lines parallel to 
top slope, through g v g 2 , ... , the true 
thrusts, g x t v g 2 t 2 , . . . , g 7 t 7 , . . . , the 

directions of gt v gt 2 , . . . , gt 7 , . . . , of 

the true thrusts on the planes A\, A2 y 

. . . , Al, . . . , all necessarily lie above 

the first assumed directions; so that the 


actual thrusts on all planes other than A5 
(which we shall regard as the plane of 
rupture, for convenience), lying above or 
below A5, make less angles with the 
normals to those planes than the angle of 
friction, just as we found in Art. 25. 

The conditions of stability of Art. 21 are 
thus satisfied in the present case; but it is 
evident that this is no longer so if we lower 
the direction of the thrust on AO, which 
lessens the horizontal component of the 
thrust from the right, since intersections 
like 6', 7', in the right diagram move towards 
the vertical Ag, though the reverse obtains 
for the diagram to the left, which of itself 
indicates some absurdity. If, now, we 
combine the new thrust on AO from the 
right (which has a less horizontal compo¬ 
nent than before) with the wedges of earth 
lying to the left of AO, it is readily seen 
that the directions of some of the resultants, 
as g f 5 , . . . , will fall below their first 
positions, and will thus make greater angles 
with the normals to their planes than the 
laws of stability will admit of; so that any 
lowering of the first assumed position, 


parallel to the top slope, of the thrust on 
AO , is impossible. 

We thus reduce to an absurdity every 
other case but the one assumed, which is 
therefore true; so that the proposition 
enunciated at the beginning of this article 
is demonstrated. 

We see, therefore, that we cannot, as 
before, assume the direction of the thrust 
on the wall, AU , as having the direction 
gm c , making the angle cf> with the normal 
to AU, and find the wedge of maximum 
thrust corresponding; but that its true 
direction, gt c , is found by combining the 
thrust found on AO, acting parallel to top 
slope, with the weight of the wedge of 
earth, OAC, between the wall and the 
vertical plane AO ; otherwise, if the left 
diagram is constructed, we find its direction 
and amount in a similar manner to that 
used in finding the direction, etc., of gL, 
. . . , by laying off on gA (produced if 
necessary) 0C X 4 ; from the end of this 
line we draw a parallel to the top slope OU 
to intersection t c , with the vertical through 
i 6 . The line ~gt c to the last scale used mul- 



58 


tiplied by |ep (where p is the perpendicular 
from A on 0 C to the scale used in laying off 
UU) gives the thrust E against the wall in 
pounds. It is laid off in position by drawing 
a line parallel to gt c through a point on AL\ 
\AJJ above A, as previously enunciated. 

29 .(</>'< <£). In case this construction 
gives a thrust on the wall which makes a 
greater angle with its normal than the 
co-efficient of friction, </>' of wall on earth, 

( f >' being less than </>, then it is correct to 
assume the direction of E as making this 
angle <£' with the normal, and proceed as in 
Fig. 4 to find the thrust. In the preceding 
article, no trial-thrust on the vertical plane 
ylO was assumed to lie nearer the horizontal 
than the top slope, as there was no reason 
for considering such exceptions to the usual 
direction in a mass of unlimited extent. 
Now, however, the wall requires the thrust 
on AO to lie nearer the horizontal than 0 C 
does, in which case the horizontal component 
will be increased (since intersections like 
7', 8', move away from the vertical Ag) y 
and the thrusts on all planes Al, A2, . . . , 
lying to the left of Ag, will be raised above 





their previous positions, gt v gt. 2 , . . . ; so 
that the thrusts on all the planes but AC 
now make less angles than <£ with the 
normals to those planes, so that the con¬ 
ditions for stability of “the granular mass ” 
are assured. 

30. The “limiting plane,” corresponding 
to the plane of rupture on the left, can be 
found by a different construction from that 
given above. Thus, having found the line 
cc 7 representing the maximum thrust from 
the earth to the right of AO, multiply by 4, 
say, and combine with the successive wedges 
of earth lying to the left of AO, on magni¬ 
fying the lines 01, 02, . . . , in the same 
proportion, thus giving the lines gt v gt v 
. . . , for the direction of the thrusts on 
the planes Al, ^42, . . . ; these all lie above 
the directions gm v gm. 2 , . . . , making the 
angles (f) with the normals to the planes, 
except for the limiting plane, where gl h and 
gm, nearly coincide, as they should exactly 
if Ah was the limiting plane. The lowest 
relative position of gt with respect to gm is, 
of course, the one selected. It is evident, 
though, that the construction for the wedge 


GO 


of greatest thrust to the left of Ay gives 
a more accurate evaluation of the thrust 
than the one to the right; so that we can 
preferably use the left construction not 
only for getting the limiting plane, but for 
finding the thrust on any wall lying below 
the limiting plane. 

It is evident, from what precedes, tliat 
the double construction of Art. 28 applies 
only when the thrust on AO is parallel to 
the top slope ; for the moment it is lowered, 
there results several planes of rupture to 
the left of AO, which is impossible. Even 
if we attempt the left construction, we have 
seen besides that the resulting thrust on AO 
is greater than by the construction on the 
right. 

In case the face of the wall, AU, lies 
above the “limiting plane,” as found 
before, we evaluate the thrust on it, as in 
Fig. 4, by assuming its direction to make 
an angle with the normal equal to (/> or to 
ft when (f>' < (f). Thus, if the inner face of 
the wall had the position A2, to the left of 
AO, the direction of the thrust on it would 
now be the gm 2 , in place of gt 2 as before. 


61 


and the conditions of stability of the granular 
mass will be found to be everywhere verified 
as in Fig. 4 (see Art. 25). 

31. Summary. — For all cases of top 
slope, when the inner face of the wall is 
battered, we first find the limiting plane by 
the construction of Art. 28 ; then when the 
inner face of the wall makes a less angle 
with the vertical than the limiting plane does 
(as is nearly always the case in practice, 
unless the surface of the earth slopes at or 
near the angle of repose, in which case the 
limiting plane is at or very near the vertical), 
we assume the direction of the thrust on it 
as making the angle <£ or (for <£' < (f>) with 
the normal, and proceed as in Art. 23, et 
seq .; but, if the face of the wall lies below 
the limiting plane, we proceed as in Art. 28, 
or if cf)' < </> we may have to proceed as in 
Art. 29, to find the true thrust. 

If the wall leans backward, there is no 
need to find the limiting plane, as the usual 
construction applies. 

For earth level at top, the l imit ing plane 
is inclined to the left of the vertical equally. 
w ith the plane of rupture to the right; as 








the top slope increases, it approaches the 
ve rtical, and coincides with it for the surface 
sloping; at the angle of re po se. 

Remark. — It is found from the con¬ 
struction to the right of Ag , in Fig. 5, for 
planes of rupture lying 7° to 14° above 
the one corresponding to the greatest thrust, 
that the thrust is less only by from 6 to 16 
per cent, though it more rapidly diminishes 
as the assumed plane of rupture nears the 
vertical. It must not be inferred, then, 
particularly for steep surface slopes, that 
a considerable divergence between the 
theoretical and actual surfaces of rupture 
will invalidate the theory, if the object is 
simply to get the thrust within a few per 
cent of the truth, particularly as the theory 
neglects cohesion. In fact, for a surface 
slope equal to the angle of repose, the plane 
of rupture is parallel to the surface ; but a 
plane lying much nearer the vertical will 
give nearly the same thrust. 

32. In this connection, it may be well to describe 
an experiment made by Lieut.-Col. Aude in 1848, 
and repeated subsequently by Gen. Ardant, M. 
Curie, and M. Gobin, on a peculiar retaining-wall 











I 




























63 


made of a triangular block or frame, in which 
the inner face was inclined to the horizontal at the 
angle of repose of the sand backing, when, of 
course, by the usual assumption as to E making 
an angle of 0 with the normal to the wall, the 
direction of E would be vertical, and there should 
be no horizontal thrust! This seemed, to the 
French experimenters, to offer a puzzling objection 
to theory; but the solution is clearly as indicated 
in Art. 28 (see Art. 67, Exps. 9 and 10). Scheffler 
indicated the correct solution as far back as 1857, 
but gave the wrong reason for it; viz., that the 
horizontal thrust was thereby greater. 

The writer, in “ Van Nostrand’s Magazine” for 
February, 1882 (p. 99), pointed out that any other 
solution than that indicated in Art. 28 was 
inconsistent with the stability of a granular mass, 
and the computations upon that basis agreed very 
closely with the experiments. Later M. Boussinesq 
has developed the theory of the limiting plane in 
connection with the attempt to complete the 
Rankine theory, by considering the influence of 
the wall on the pressures even to a finite distance 
from it. According to Flamant, he defines two 
limits to the thrust, and considers the most probable 
value the smaller of these limits augmented by 
2 9 2 - of their difference. From an examination 
of the numerical values computed by Flamant 
(“Annales des Pouts et Chausses,” April, 1885), 
the results do not differ greatly from those given 
by the simple theory alone used in this work. 


G4 


33. The disturbing influence of the wall in 
changing the normal character of the stresses can 
be illustrated as follows: If the thrust on the 
vertical plane ^fO (Fig. 5) acts parallel to the 
surface 0 — 10, it meets the plane of rupture at 
one-third of its length above A, through which 
point the weight of the prism of rupture acts 
also; so that the resultant on this plane acts at 
this point, which corresponds to a pressure on 
the plane of rupture uniformly increasing from the 
surface downwards. If, however, the wall causes 
_ ( ) 

the thrust on AO to make a{ ‘" . angle with 

( greater ) 

the horizontal, the resultant on the corresponding 

plane of rupture on the right acts j j the 

( above ) 

point situated at one-third of the length of the 
plane above A, so that the pressures on it are no 
longer uniformly increasing. This abnormal state, 
doubtless, does not extend far into the mass before 
the usual direction of the thrust in a large mass of 
earth is attained; but the fact throws doubt upon 
the assumption of a plane surface of rupture for 
all cases where the direction of the thrust on the 
vertical plane does not act parallel to the upper 
surface. 

It appears reasonable to suppose, if the line 
through the centres of pressure on all sections 
of a retaining-wall passes through their centres 
of gravity, that no rotation of the wall occurs; 
further, if it was possible for the masonry and 


f 



earth or rock backing to settle together the same 
amount, — the backing, say, having been carefully 
deposited in horizontal layers,—then, even fora 
level-topped bank, the maximum thrust, as given 
by Rankine’s formula, will be exerted, and there 
will be no friction at the back of the wall to change 
the usual direction of the earth-thrust in a large 
bank. If the wall has not the stability, or the 
settling is not as assumed, the top will move over, 
friction at the back of the wall is exerted, and the 
horizontal thrust becomes smaller than before, 
corresponding to a different prism of thrust, as 
we ascertain by the construction of Fig. 4, for 
the two cases of E horizontal and E inclined 
downwards. The excess of the horizontal thrust 
in one case over that in the other must necessarily 
be resisted by the ground-surface, on which the 
filling rests by friction, which it is generally 
capable of doing. If not, then the Rankine 
thrust will be exerted. Similarly, if we consider 
any road embankment, whose sides slope at the 
angle of repose, the horizontal thrust on some 
longitudinal plane in the interior must be finally 
resisted by the ground to one side under the 
embankment. If, however, the weight of earth 
above, multiplied by the co-efficient of friction of 
earth on ground-surface, is less than the horizontal 
thrust, the earth must slide, and the slope become 
flatter, until equilibrium obtains from a less 
horizontal thrust. Scheffier computes for an 
embankment of triangular section where 9 = 45°, 


GO 


and the angle of friction on the ground-surface is 
only 5°, that the slope of the embankment would 
change to 32° 15'. For the ground-friction angle 
= 7° 7' 20" there would be exact equilibrium; so 
that, generally, there need be no fear from spreading 
of embankments due to this cause, as the amount 
of friction required is very small. 

34. We have now given methods for 
finding the thrust against a retaining-wall, 
which simply resists this active thrust of the 
earth, for the usual cases of a surcharged 
wall and earth-level at the top, to which 
may be added the case of earth sloping 
downwards from the top of the wall to the 
rear, for which the construction is evident. 
It now remains to find the passive resistance 
of the earth to sliding up some inclined plane 
due to an active thrust of the wall from left 
to right (Fig. 4), caused by water, earth, 
or any other agency acting against the wall 
on the left. Now' (Art. 22) we la} 7 off the 
angle hgs (Fig. 4) above bg , and then, from 
the new position of s. lay off ares ss v ss 2 , 
. . . , below s equal to ga v ga. 2 , . . . , as 
before, giving the direction of gs v gs 2 , . . . , 
inclined at angle </> above the normal to the 


corresponding planes Al, A’2 , . . . The 
construction then proceeds as before, only 
it is now the least of the resistances, gc, that 
represents the passive resistance of the 
earth to sliding up the plane of rupture 
corresponding ; for any increase over this 
causes the thrust on some planes to make 
greater angles than cf> with the normals, as 
is easily shown. It is plain that gc has 
the same direction as before; for if we 
call N the component normal to AC of the 
resistance, since, if the earth moves upwards 
along the plane of rupture and plane AC, 
the friction of the earth along AC, JSf 
tan. cf >, acts downwards, which gives the 
direction of E, representing the earth- 
resistance, inclined below the normal as 
before at the angle c p, the same as for the 
active thrust of earth on wall. This active 
thrust is of course the only one exerted, 
unless the wall tends to slide, so that the 
consideration of the passive resistance is 
of small practical value. In case of a 
heavy structure resting on a foundation, we 
can replace the total weight b}- that of earth, 
and estimate the active thrust exerted against 


n vertical plane just below the foundation, 
for the full weight of the supposed earth, 
by the method to be given in the next 
article. The earth to one side of this 
vertical plane can be conceived to exert a 
passive thrust, which may be estimated as 
explained, and should exceed the active 
thrust for a stable foundation. This 
method, though, of estimating the stability 
of a foundation, while doubtless on the 
safe side, is otherwise illusory, as any one 
who lias seen a heavy locomotive move at 
great speed along a narrow embankment 
must admit. The mass, by its friction, 
rapidly and safely transmits and distributes 
the weight over the ground, without exerting 
any horizontal thrust at the side slopes, 
which are perfectly stable. 

3f>. Underground Pressures. — To find 
the unit pressure at a depth x below the 
surface of a large mass of earth, level at 
top, of indefinite extent, and resting upon 
a uniformly compressible foundation, every¬ 
where at the same depth (see Art. 9), we 
proceed as follows: Let Fig. G represent a 
slice of the earth contained between two 


CO 


vertical planes one unit apart, and bounded 
on one side by the horizontal plane 0C y , at 
a depth x below the surface, on the left 
by the vertical plane ^10, whose depth is 



Acc, and below by the plane AC; the 
planes AO, OC , and AC being supposed 
perpendicular to the plane of the paper. 
Let the length AO = Ax, and the length 
OC = n. Ax. The plane AC will be con- 























70 


sidered to take successively the positions 
Al, A2 , . . . ; so that if we divide 
ylO = Ax into ten equal parts, as shown, 
and lay off similar equal parts along 0C\ 
as AC varies in position, n will take the 
successive values 0.1, 0.2, . . . Calling 
e the weight per cubic foot of earth, the 
weight of the prism of earth resting verti¬ 
cally over 0 C is represented generally by 
e.x.n . A.i*, which, beingdirecttyproportional 
to w, we can lay off on the vertical 0^4 the 
lengths 01, 02, . . . , to represent the suc¬ 
cessive values of ?i, or the vertical loads 
sustained by the horizontal bases 01, 02* 
. . . , of the successive prisms considered. 
When the length Ax is very small, we can 
neglect the weight of the small prism of 
thrust, x40C, in comparison with the weight 
of the vertical prism above it, without 
appreciable error, and ultimately find the 
position of the plane AC, which gives the 
true thrust against ^40, by previous methods. 

Thus, draw the quadrants shown with A 
and 0 as centres, and ^40 as a radius ; note 
the intersections a v ct 2 , . . . , of the lines 
Al, A'2, . . . , with the arc 0D ; next, 





71 


construct angle Cos = c/> the angle of 
repose of the earth, and ares ss, = 0 
ss 2 = 0a 2 , . . . ; so that each of the lines 
OSj, 0s 2 , . . . , next drawn, make the angle 
cf) with the normals to the corresponding 
planes >41, >42, . . . , and thus represent 
the direction of the resistances offered by 
these planes in turn regarded as planes of 
rupture. On drawing horizontals through 
the points of division 1,2,. . . , on AO to 
intersection 1", 2", . . . , with the cor¬ 
responding directions 0s 1 , 0s 2 , . . . , we 
note, that, if the thrust on AO is taken as 
horizontal (Art. 9), the lines 11", 22", 
. . . , represent the horizontal thrusts 
caused by the weights resting on the suc¬ 
cessive prisms >401, >402, . . . , treated as 
successive wedges of rupture. The greatest 
of these 77" represents the actual thrust on 
AO ; for if we assert that any other, as 44", 
represents the actual thrust, to get the 
corresponding thrusts on all the planes >41, 
>42, . . . , in direction and amount, we 
must lay off a length equal to 44" along 
each of the horizontals 11", 22", . . . , 
produced if necessary, and through the 


extremities draw lines to 0, which thus 
represent in amount and direction the 
thrusts on the corresponding planes. But 
since 44" is less than T7 77 , this construction 
will give a thrust on the plane -47, lying 
below the position 07 77 , and thus making a 
greater angle than cf> with its normal, which 
is inconsistent with the laws of stability of 
a granular mass. Hence, any other thrust 
than the maximum, as given by the above 
construction, is impossible. 

The length of 77" to scale is 0.52, which 
we must now multiply by ex tlx to get the 
total horizontal thrust on the plane .40 in 
pounds. On dividing this thrust (0.52 
ex Ax) by the area pressed = 1 X Ax, we 
get the unit pressure on a vertical plane 
at a depth x below’ the surface equal to 
0.52 e . x, which is called “the intensity of 
pressure,” at a depth x. As w r e neglected 
the weight of the prism 40 C, we must 
conceive Ax to diminish indefinitely, so that 
the error tends indefinitely towards zero, 
and the approximate intensity of pressure 
on A0 = Ax approaches indefinitely that 
at the point 0. 






By analysis we shall show hereafter that 
the plane of rupture, A'l in this case, bisects 
the angle between the natural slope and the 
vertical. 


In this case we have taken 0 = 18° 26\ and the 
resulting intensity (0.52ex) is found to be exactly 


that given by the usual formula, ex tan 2 



The intensity at any point of a vertical plane thus 
varies directly with x. The total amount on a 
vertical plane of depth x from the surface is then 



Cx 2 

(where C = 0.52e in the present 


case), and its resultant is at a depth z equal to the 
limit of the sum of the moments of the pressures 
( Cxdx ) on the elementary areas dx X 1, taken about 
the top surface, divided by the total pressure, or 

z = 


r x Cx 2 

I Cxhlx ~ 

Jo 2 


= -x. 

o 


A . Cx L ex a '■'o — ex w i. . 

Also, -7- = — X O.o2x = — X line represent- 
2 2 2 

ing thrust, if old construction is used. These 
are precisely the conclusions derived from previous 
constructions. 


In case the top surface is sloping, a 
similar construction applies, only 0(J must 
now be drawn parallel to the top slope, and 
the pressure on OA must be assumed to act 







74 


parallel to this direction. The construction 
is similar to that given for Fig. 5 (on 
neglecting the weight of the wedge of 
thrust as above), either to the right or left 
of the vertical Ag, only as the weight of 
the prisms vertically above 01, 02, . . . , 
(Fig. 5) is now represented by ex nAx x 
cos i (where i is the inclination of the top 
slope to the horizontal), we must multiply 
the length of the line cc' (Fig. 5) to scale, 
b y ex cos i, to get the intensity of the 
pressure at the depth x , since the lengths n 
alone were laid off to represent the loads 
gg v yg v . . . , as in Fig G, and the resulting 
thrust cc' must now be magnified ex Ax cos i 
times to get the thrust in pounds on the 
plane Ax x 1. As Ax approaches zero 
indefinitely, the approximate intensity 
ex Ax cos i —, 

- cc , on the area Ax x 1, ap- 

Ax 1 

proaches that at the depth x (= ex cos i.cc') 

as near as we please. It must be observed 

that AO in Fig. 5 must be taken equal to 

unity in this construction, and the same 

scale used in laying off the distances along 

the top-slope 0 — 10. 



75 


36. If the earth to the right of MO, in 
Fig. 6, does not experience the similar 
active thrust of earth to the left of MO, but 
only the i^assive resistance of a tunnel 
lining, etc., of an underground structure, 
the conditions are changed if this lining 
gives in consequence of its elasticity; for 
the wedge of thrust, AOC , cannot move 
to the left without developing friction along 
the surface 0(7, therefore the pressure on 
this surface must no longer be taken as 
vertical, but as inclined at a direction 
0 — 10', making an angle </> with the vertical 
(Fig. 6)'. The load on any supposed wedge 
of thrust, as M04, is now represented by 
04', the thrust on A0 by 4 / 4 // , and the 
pressure on the plane M4 by 04". The 
greatest of the lines, l'l", 2'2", . . . , now 
represents the true thrust, and it is readily 
found to be 4'4" = .33 to scale ; so that the 
intensity of the thrust on a square foot at 
the depth x is now 0.33ex, or one-third the 
intensity on the horizontal plane 06'. Mr. 
Baker (“Science Series,” No. 56) found 
for a heading, driven for the Campden-Hill 
Tunnel, at a depth of 44 feet from the 





76 


surface, —the angle of repose of the over- 
lying clay, sand, and ballast, heavily 
charged with water, being only 18° 26' as 
assumed above, — that the relative deflec¬ 
tions of the timbering in the roof and sides 
indicated that the vertical and horizontal 
intensities of pressure were in the ratio of 
3.5 to 1, which is very near what we 
obtain by the last construction. The first 
construction indicates a ratio of only 2 
to 1. 

In most cases, a portion of the weight of 
the earth above the tunnel is transferred 
to the sides (Art. 9), though here it was 
thought that “the full weight of the ground 
took effect upon the settings.” 

We have now carefully examined the 
conditions of interior equilibrium of a mass 
of earth, and ascertained the thrusts exerted, 
whether in the interior or against a retaining- 
wall; and we see that the graphical method 
is capable of handling, with equal ease, any 
case that ordinarily presents itself. The 
results, of course, agree with the analytical 
method, founded on the same hypotheses ; 
but as it is often more convenient to cal- 


77 


culate the thrust, eveu when a graphical 
method is afterwards used for testing the 
stability of the wall, we shall now proceed 
to deduce formulas for evaluating it. 


CHAPTER III. 


THEORY OF RETAINING-WALLS. 

Analytical Method. 

37. As in the preceding chapter, we shall 
assume a plane surface of rupture, and re¬ 
gard the mass as subject only to the laws 
of gravity and frictional stability stated in 
Art. 21. 

In Fig. 7. let AFPQRC represent a cross- 
section of the earth-filling, taken at right 
angles to the inner face of the wall AF. 
We shall consider the conditions of equi¬ 
librium of a prism of this earth contained 
between two parallel planes, perpendicular 
to the inner face of the wall, and one unit 
apart, regarding the wall AF as resisting 
the tendency of the earth to slide down 
some plane, as AC , passing through its 
inner toe. 

Call G the weight of the prism of earth 


78 


79 


AFPQRC in pounds, directed vertically; 
E , the earth-thrust against the wall AF y 
directed at an angle </>' of friction of earth 
on wall when c f> < <£, or of <f> when <f> > <£, 



below the normal to the inner face of the 
wall (Art. 7) ; and S the reaction of the 
plane AC, inclined at an angle </> (the angle 
of repose of earth) below the corresponding* 
normal, since the prism is supposed to be 
on the point of moving down the plane 









80 


AC. These three forces fire in equilibrium 
when E and S act towards 0, and G acts 
downwards. 

Call the angle that AC makes with the 
horizontal y, and the angle FAC , /3. On 
drawing the parallelogram of forces as 
shown, we have, since E and G are pro¬ 
portional to the sines of the opposite angles 
in the triangle ONL , 

E = sin ONL 
G sin NLO 

It is easily seen from the figure that ONL = 
y — <£, and that NLO = cf) -f- /3 -f- <£'; 
lienee the above general relation becomes, 

E = sin (y - cft) m 

G sin (</> + <£' + P) ' * ’ K 

Now, if we conceive the plane AC, always 
passing through the point A, to vary its 
position, that value of E, corresponding to 
the greatest value obtained by the construe- 
tion above, is the thrust actually exerted 
against the wall; for, if ^lC y is the plane of 
rupture corresponding to this greatest trial 
thrust, any less value of the resistance of 




81 


the wall E will cause S to make an angle 
greater than cj) with the normal to AC , 
which (Art. 21) is inconsistent with the 
law of stability of a granular mass (also 
see Art. 25) : hence, the least thrust con¬ 
sistent with equilibrium corresponds to the 
greatest value of E thus obtained ; and this 
is the actual active thrust exerted against 
the wall, when the wall simply resists the 
tendency to overturning or sliding on its 
base, caused by the tendency of the prism 
of rupture to descend. If there is a thrust 
exerted on the wall towards the earth, from 
any external force acting on the left of the 
wall, from left to right; then, if this be 
supposed to increase gradually, the active 
thrust of the earth on the right is first 
overcome; then, as the external force 
increases, the directions of S , on all planes 
as AC , approach the normals to those 
planes, pass them, and finally the full passive 
resistance of some prism of earth to sliding 
upwards along its base is brought into 
play. The greatest resistance, E , to 
sliding up the base of some prism which 
can be exerted is that corresponding to the 


82 


least of the trial resistances, E , obtained 
by supposing the position of the plane AC 
to vary, for S lying above the normal to 
AC at an angle </> for each plane ; for if we 
suppose AC to represent the corresponding 
plane of rupture, if the external force, equal 
to E , and acting from left to right, is in¬ 
creased, it necessarily causes the direction 
of S to make a greater angle than (f) with 
the corresponding normal, which is incon¬ 
sistent with equilibrium (Art. 21). 

In this chapter we shall only consider 
the passive resistance of the wall to over¬ 
turning or sliding caused by the active 
thrust of the earth tending to descend, 
which is all that is required in estimating 
the stability of retaining-walls. 

38. We shall now express the value of 
G for the earth-profile shown in Fig. 8, 
taken to represent the general case, and 
proceed to find the maximum value of E , 
for different trial-planes, which represents 
the actual thrust exerted against a stable 
wall. We shall suppose the true plane of 
rupture to intersect the part of the 
profile; the line li Y is then produced to 


83 


B , so that the area of the triangle ABC 
is equal to that of the polygon AFPQRC , 
which can be effected by ordinary geomet¬ 
rical means. The point B therefore does 
not change, as we suppose the position of 
C to vary between R and Y. 



Let us drop the perpendicular AT from 
A upon BY, and designating by e the 
weight per cubic foot of earth, we have 
G = \e.AT.BC. 

For future convenience we have desig¬ 
nated, in Fig. 8, the angle that AC makes 
with the vertical w, and the angle that the 
inner face of the wall AF makes with the 









84 


vertical a; so that the angle /3 of (1) is 
now replaced by (w -f a) if the wall leans 
forward, or by (w — a) if the wall leans 
backwards. 

In Fig. 8, let us draw the line Cl , mak¬ 
ing the angle AC I = (<£ + cf)' ■+• /?) = 
(cf) + cf)' -f* <o -f- a) to intersection (7, with 
the line of natural slope AI) through A. 
If the wall leans backwards, 

AC I = (<p + f -)- w — a). 

Since the angle (y — cf>) = CAI , we can 
replace the ratio of the sines in (1) by that 
of the sides opposite in the triangle ACI , 
or of Cl to AI; so that, substituting the 
above value of (7, we can write (1) in the 
following form : — 

E = * e.AT.BC. — . . . (2). 

AI v ’ 


On drawing BO j^ircdlel to Cl to inter¬ 
section 0 with All we have, from this 
relation and the similar triangles, BOD 
and CID , 


BC = 


BD. 


qi . 

OD’ 


Cl = ID. 


BO. 

OD’ 





8o 


which substituted in (2) gives, 


E = 


—6 

‘2° 


' AT.BD.BO 


0D“ 


> 


OLID 

AI 



The terms in the ( ) remain constant as 
we vary the position of AC. For brevity, 
call AI = as, AD = «, AO = b; then we 
can write the variable term, 

OLID _ (x— b) (a — x) _ t , u ((b 

—— — — Cl -f- 0 X, 

AI x x 

which is a maximum for x — \ab, as we 
find by placing its first derivative equal to 
zero. This value of x substituted in the 
variable term gives, 

a + b- Hab = < a ~ 

a 


so that the actual thrust E on the wall can 
be written, — 

J S = ie (AT.B D . B q ya- _ ^ b r _ (4)- 

' V oif 1 « 

Now, drawing the perpendiculars BN and 
CH from B and C upon AI), we observe 
that since the angle ACH = o> -f- <£ (AC 
makes the angle a> with a vertical at (7, 











86 


and CII makes the angle (f) with this same 
vertical, since the sides are respectively per¬ 
pendicular to those of the angle DAJ = <£), 
and the whole angle AC I = (<d 4- -Ta), 

it follows that the angle IICI = NBO — 
(V + «) as marked, if the wall leans 
forwards ; otherwise IICI=NBO = (<(>' — a), 
since ACI is then equal to cf>' — a), 

as previously observed. 

To reduce (4) to a simpler form, we 
remark that AT.BD represents double 
the area of the triangle ABD , and can be 
replaced by AD.BN — AD.BO cos OBN; 
which gives 


j 


1 T.BD.BO 


Gif 


— a cos OBN 


E = ie. cos OBN(~S 

\0D) 




Now, from similar triangles, BOD , CID , 
we have which, substituted in 

the above expression, and this in turn in 
(4), we have, noting that (a — \/ab) = 
(a — x) = JZ>, the very simple formula, 






87 


E = \e. cos (<// -f- a) CP ... (6). 

It is to be remarked, that, if the wall leans 
backwards, cos (</>'-}- a) is to be replaced 
in this formula by cos ((f)' — a) ; further, 
if we lay off IL = IC on the line A4, and 
draw a line from L to C, the thrust E is 
exactly represented by the area of the 
triangle ICL multiplied by e, the weight 
per cubic foot of the earth. 

39. This simple conclusion has been 
previously reached, in an entirely different 
manner, by Weyrauch (see “Van Nostrand’s 
Magazine” for April, 1880, p. 270), who 
states that Rebhahn in 1871 found a similar 
result, assuming, however, that <f>' = 0, or 
cf>' = <f) (for the special cases of earth-level 
at top, or sloping at the angle of repose, I 
infer). 

Recurring now to the fact, that for the 
true plane of rupture we found 

x = AI = Sjab = V AD.AO , 

and that angle NBO = (cf>' + «) or (</>' — a), 
according as the wall leans forwards or 
backwards, we have the following simple 
construction to find the plane of rupture 



88 


and earth-thrust E , as given by Weyrauch 
in 1878, for a uniform slope and wall lean¬ 
ing forward. 

Having found the point B on the pro¬ 
longation of the line R Y, which it is thought 
will be intersected by the plane of rupture, 
so that area ABR = area AFPQR, we 
next draw BO, making the angle NBO 
with the normal to the line of natural slope 
AD, equal to ( cjI + a) or (<// — a), ac¬ 
cording; as the inner face of the wall lies to 
the left or to the right of the vertical through 
A (replace </>' by (f> whenever cf>' > </>) ; then 
erect a perpendicular at 0 to AD to inter¬ 
section M, with the semicircle described 
upon HD as a diameter, and lay off AI — 
chord AM, since AI = sjAO.AD; next, 
draw IC parallel to OB to intersection C 
with the top slope, whence AC will be the 
plane of rupture if the point C falls upon 
R Y as assumed ; otherwise another plane, 
as YZ, will have to be assumed as con¬ 
taining the point C, and the construction 
effected as before. 

Having found C in this manner, E can 
be computed from (1), since G = ^ AT.BC 



89 


is now known ; or by measuring Cl to scale y 
E can be found directly from (6). 

This graphical construction is more rapid 
and accurate in working than the methods 
of the preceding chapter, and is superior to 
Poncelet’s construction, in taking less space 
to effect. 

In surcharged walls, the point B will 
generally lie to the right of AF. Thus, in 
Fig. 4 the upper line 26 is extended to 
the left; from 0 a line is then drawn parallel 
to A'2 to intersection O' with the line 26 
extended. The point 0' thus found corre¬ 
sponds to the point B of Fig. 8. 

40. The construction is true whether the 
earth-surface slopes upwards or downwards 
from the top of the wall. 

In the latter case, if the surface, say BD , 
falls upon the line BO, the construction fails ; 
but a formula given farther on gives the 
value of E. 

If the surface BD falls below BO , it is 
easily seen, on drawing a figure, that all the 
previous equations hold, and we reach the 
same conclusion as before, AI = ^AD.AO ; 
only as AO now is larger than AD , the 



90 


semicircle must be described upon AO as 
a diameter, and the perpendicular to the 
point M erected at D ; or AI can be calcu¬ 
lated if preferred. If the points O, /, 
and D are near together, it will be best to 

compute BC from BC = BD .——, since 
1 OD 

the terms in the right member can be meas¬ 
ured to scale. 

41. Position of the Limiting Plane. —In 
Fig. 9, let BD represent the earth-surface, 
uniformly sloping at the angle i to the 
horizontal, of an unlimited mass of earth 
(Art. 9), in which the pressure on a vertical 
plane, AB, can be taken as parallel to the 
surface BD. Let AD represent the line 
of natural slope ; it is required to find the 
position of the plane of rupture AC, corre¬ 
sponding to the thrust E, acting above the 
horizontal at the angle i, and of course 
balancing the opposed thrust of the earth 
to the left of AB. 

On referring to Fig. 7, it is seen that equa¬ 
tion (1) holds on replacing the denominator 
of the right member by sin (/? -f- d) — i). 
Therefore, in Fig. 8, the angle AC I must 



now be laid off equal to (# + <£> — i) r 
whence the line Cl falls below CII, and 
BO below BN , botli being; inclined to these 
normals at the same angle, 0'+ a = i -1- 0 = 
With this exception, the above demon¬ 
stration holds throughout, and we reach the 



following construction to find the point (X 
From B draw BO , making the angle i below 
the normal BN to AD, or preferably making 
the angle (<£ — i) with the vertical AB , to 
intersection O with AD. From 0 draw 
OM perpendicular to AD to intersection M y 
with the semicircle described upon AD as a 






92 


diameter; lay off AI along AD, equal to 
-chord AM, and from I draw a parallel to 
BO to intersection C with the top slope 
BD. The plane A C is the plane of 
rupture, or the limiting plane of Art. 28, 
which see. 

If the inner face of the wall lies below 
AC, then (Art. 28) the thrust=^e. eos i. Cl 2 
on AB is computed, and, regarded as acting 
parallel to BD, from left to right, is com¬ 
bined with the weight of the earth and wall 
to the right of AB to find the true resultant 

•ZD 

on the base of the wall. 

If the wall lies between AB and AC, the 
constructions of Arts. 37 and 38 are used. 

To be as accurate as possible in these, as 
in all constructions, true straight edges on 
both ruler and triangle are imperative. 
Lay off all angles, including right angles, 
by aid of a beam compass to a large radius, 
say ten inches, using a table of chords (ex- 
cept for the right angle) and an accurate 
linear scale. With all care, the angles 
BAG thus found can scarcely be counted 
on to nearer than ten minutes, which, how¬ 
ever, is sufficiently accurate. 


93 


In the table below will be found, for 
various inclinations i, the values of the 
angle BAC that the limiting plane makes 
with the vertical; also the co-efficient K 
(see Art. 42), or the thrust on AB = \e 
cos i CP, when AB and e are both taken 
as unity, made out for earth which naturally 
takes a slope ot‘ one and a half to one, or 
whose angle of repose is 33° 42'. 

The value of K agrees fairly well with 
calculation, the last figure not differing 
more than one or two, at the outside, from 
the results of Art. 47. 

From the construction we see that as i 
approaches cf) indefinitely, BAC tends to zero 
and E approaches the limit \e cos $. AB 1 , 
as given by analysis. The increase of thrust 
is very rapid from i =30° to i =(f> = 33° 42'. 


i 

0° 

5° 

10° 

15° 

O 

O 

i 

25° 

26° 34' 

CO 

o 

o 

33° 42' 

BAC 

28° 09' 

26° 

24° 

21° 50' 

19° 10' 

16° 

14° 40' 

11° 10' 

0° 

K 

.143 

.145 

.149 

.157 

.172 

.194 

.207 

.244 

.416 




























94 


42. Uniform Top Slope; Formula for 
Earth-thrust. — When the upper surface of 
the earth slopes uniformly at the angle i to 
the horizontal, it is easy to deduce from 
what precedes a general formula for the 
thrust exerted by it. Fig. 10 represents a 



retaining-wall leaning towards the earth. 
We shall first deduce a formula for this 
case, when it will be observed, as we pro¬ 
ceed, that the same formula holds, when 
the wall leans forward, on simply changing 
a to (— a). 

In this case, we note from Fig. 10 the 
following values for the angles : — 






95 


NBO —(f)' — a, 

ABO = $ + <//, 

AOB = 90 - (<// - a), 
ylZhB = <£ - i, 

ABD — 90 —j— a —j— 2, 
BAO =90 -(</> + a). 


Finally, designate by l the length AB from 
the inner toe to where the inner face of the 
wall intersects the top slope, and by li its 
corresponding vertical projection. 

From formula (5) we deduce, remem¬ 
bering that OD = (a — 6), 

— V ab 12 


E = be. BO 1 


a 


a — b _ 


. cos OBN . . . (7). 


We can now write the [ ] as follows : — 


a 


— \Jab _ 1 



a — b 



a 



Place n = 



to find its value in terms 


of the functions of known angles, we have 
from the triangles AOB and ABD by the 
law of sines, 









96 


AO __ sin (<f) + 4>') AB _ sin (0 — a) 
AB cos ((f)' — a)’ AD cos (a -f- i ) 

On multiplying these two equations to¬ 
gether,, and extracting the square root, we 
find, 

, :=;t /^n7 = ./ sin (4>+4>') si » (<£-*) _ ( 8 v 

\ V COS ((f)' —a) COS (o + f)’ 

Again, from the triangle 22 CM, we have, 
BO = cos (ft + a ) Z. 

COS (</)'— a) 


Substituting these values in (7), and putting 
cos OBN = cos (cf)' — a) for this case, and 
we have finally, 


E 


/ cos ( (j) + a) 


-( 


) = 


el 2 ' 


n+ 1 / 2 cos ( (j)' — a) 


(9). 


Or, since h = l cos a, we likewise have. 


E _ / cos (<ft + a) 


» + «) V 

1 ) COS a/ 2 C( 


c/i 2 


1 ) COS a/ 2 COS ((f)' —a) 


. . . ( 10 ) 


If we term the co-efficient of e/i 2 in (10) 
/t, we can write this formula, 

E = Kelt 2 . . . (11) 















97 


in which, for walls leaning backwards , as in 
Fig. 10, 


K = 


/ cos ((ft + «■) 

\(tt-f-l) COS a 


o 


2 COS (</>' — a) 


••( 12 ), 


where n has the value given in (8). 

For walls leaning forwards , we easily note 
the changes in the angles used, and can 
verify that formula (11) obtains; but 
now, 


__ ^ COS ((f) a) 


(n- J-l)coSa/ 2 COS (<£'-j-a) 


(13) 


and, 





sin ((/> -f- (f)') sin (cf) — i) 
COS (^)' + a) COS (a — l) 


• ( 14 ); 


which we obtain from the old values by 
simply changing a to ( — a). 

It is to be observed, for all cases, that 
when cf)' > cf) that we must replace <// in all 
the formulas by cf). 

These formulae are identical with those of 
Bi *esse (“ Coin’s de Mecanique Appliquee,” 
Yol. I. 3d ed.) and Weyrauch, for the 
case of the wall leaning forward, the only 
cases examined by them. Bresse uses the 








t... 


98 


Poncelet method for the general case, which 
leads to Poncelet’s celebrated construction. 
The routes pursued by these authors is dif¬ 
ferent from that given above, the method 
of Weyrauch, in particular, being much 
more complicated ; still, all three methods 
lead to precisely the same formula, so that 
it must be considered as established beyond 
question. 

Weyrauch, too, in subsequent reductions, 
follows Rankine as to the direction of the 
earth-thrust against the wall, whereas Bresse 
takes it as above. The case of the “ limit¬ 
ing plane ” is not considered by either. 

43. The case where the top surface slopes 
downwards to the rear is very rarely met 
with in practice. The previous formulae 
apply though directly on simply changing i 
to (— i), since it is seen that angle ADB 
==((/> + i) and angle ABD = 90 + (a — i) y 

and the ratio 4— ls now equal to S * n (ft 0 
AD 1 cos (a - i) 

44. Earth Level at Top; Back of Wall 
Vertical. — For the earth level at top, back 
of wall vertical, and <£' = <£ as usually 
taken, the formula (11) takes a very simple 




99 


form. Here we have a = 0, <// = </>, i = 0, 
whence, 




. /sin 2 </> sin <±> . ,. /s 

= V-^-7—- = sill </>V2, 

V cos 9 


and 




COS <£ 


= 73 . eli 2 . . . (15). 


2 (1 + sin c/> V 2 ) 


For <f / = 0, which corresponds to a per- 
fectly smooth wall , or otherwise may refer 
to the direction of the pressure on a vertical 
plane in a mass of earth of indefinite extent, 
level on top (Art. 9), we have, when a = 0 
and i = 0 , n = sin 0 and, 

__ 1 — sin (ft eli 2 
1 -f sin (f> 2 

,2 (A ~o _ <£' 


= tan 2 ( 45 — 


eli 2 

T 


. . . (16). 


The equality of the two co-efficients of 


eh 2 

2 


in (16) is easily verified from the known 


formula, 


tail 


2 1 


(*) = 


1 — COS X 
1 -b COS X 


by putting (90 — </>) for x in both members. 
Referring to Fig. 7, and regarding AF 


• . 










100 




vertical, the top surface horizontal, and 
(f)' = 0, we note that O — - . h 2 tan ft and 

E = -h 2 tan ft tan (y — (/>), in which y = 
2 


90 — ft. Now, this result must agree with 
the right member of ( 1 G), which is 011 I 3 7 

possible when ft = ^45 — or 2ft = 

(90 —(p) ; whence it follows that for (f>' = 0, 
a = 0, i = 0, as assumed, the %)lane of 
rupture bisects the angle betiveen the vertical 
and the line of natural slope. 

45. Earth sloping at the Angle of Repose. 
— For this case we shall assume a =0 and 
(f' = in addition to i = <f>, whence n = 0 
and, 

E _ cos_£ _ e7(2 _ _ _ (17), 


as found in a different manner in Art. 41. 
This simple formula can likewise be 
deduced directly from equation (1) of Art. 
37, referring to Fig. 7, 

E _ sin (y — c _ cos (ft + <fr) 

G sin (<j) + <f)' + ft) sin (2</> -f- ft) 


0 


*) 





101 


On substituting the value of 6r, which is 
easily found for this case to be, 

q __1_ eli 2 _ sin (3 cos eh 2 

cot (3 — tan (f> 2 cos ((3 + (p) 2 ’ 

we find for the trial thrust, 

g _ sin /3 cos e/t 2 
sin (2</> 4- (3 ) 2 

__ cos <p _ eh? 

sin 2(f) cot /? + cos 2 cf) 2 

Now, by the reasoning of Art. 37, the 
true thrust is the greatest value the above 
expression can have, as (3 varies, and its 
greatest value corresponds to f3 = 90 — (f >; 
for then cot (3 is least, and E greatest, since 
cot (3 is in the denominator. On substitut¬ 
ing this value a simple reduction gives 
E = ^ cos </> . eli 2 as found above in (17). 
Since we have just found, for this case, 
that [3 — 90 — </>, it follows that the surface 
of rupture coincides toith the natural slope . 
The value of E from equation (1) in this 
case assumes the form Ox oo, since G 
becomes infinite for an indefinitely sloping 
surface; but on reducing to the form above 








102 


we easily see the limit that E approaches 
indefinitely, which is its true value. The 
construction of Art. 39 fails for this case, 
but the one of Art. 41 leads directly to (17). 

4G. Pressure of Fluids .—The general 
formula (9) above is true, no matter how 
small the angle of repose $ becomes, and 
must approach indefinitely the expression 
for the pressure of liquids, as </> and <£' 
tend towards zero ; so that at the limit, for 
<f) =</>' = i = 0 , we have the normal thrust 
of a liquid whose weight per cubic foot 
is e, 

E = 4 cos a _ 1 e J L 2 sec a ' _ . (18), 
a well-known formula. By Art. 44 we see 
that for cf) = 0, 2/3 = 90, or the plane of 
rupture approaches an inclination of 45° as 
<£ approaches zero indefinitely. 

47. Rankine's Formula for the Earlli- 
thrust on a Vertical Plane , in an Indefinite 
Mass , sloping uniformly. — In Art. 9 we 
have stated the conditions that such a mass 
must satisfy in order that the pressure on a 
vertical plane, whose intersection with the 
top slope is a horizontal line, may be 
parallel to the line of greatest declivity. 


103 


Also in Art. 28 we have seen, that, when 
the wall face lies below the limiting plane, 
this direction of the thrust is the true one 
on a vertical plane, passing through the 
inner toe of the wall. 

We have a = 0,</>'= i, and l = h , which 
gives in formula (9), 

_ / cos (ft \ - eh 2 
\n -f- 1 / 2 cos i 

where, 


n — 


/sin ((f) + i) sin (^> — i) 


cos - i 


_ Visin ' 2 cos ' 2 i — cos 2 sin 2 i 

COS l 


y/cos 2 i — cos 2 

cos i 


Whence, 

r, cos 2 6 cos i eh 2 

(cos j -j-Veos 2 t — cos 2 </>) 2 ^ 

Now, since we can write, 

cos 2 <f> = (cos -j- Vcos 2 i — cos 2 <£) 
X (cos i — Vcos 2 i — cos 2 </>) 


the above value becomes, on striking out 















104 


the common factor, (cos i 4 - Vcos 2 i — cos 2 <£) 




eh 2 

cos ? 


cos i — Vcos 2 i cos 2 
cos i 4 - Vcos 2 i — cos 2 </> 



which is Rankine’s well-known formula for 
earth pressure. 

Now since Rankine’s formula was framed 
without the use of any assumption, as that 
of a plane of rupture, and is accepted as 
correct for the case in question, it follows, 
that, when the pressure is assumed to be 
parallel to the surface, the assumption that 
the surface of rupture is a plane will give 
correct results, and can be safely used in 
the graphical method which is absolutely 
dependent on it. 

It will be observed that formulae (16) 
and (17) can be deduced directly from (19) 
by making i = 0 and i — (j) respectively. 
Rankine has given a simple graphical con¬ 
struction of the last fraction in (19) in his 
“Civil Engineering,” which saves labor in 
computing. 

48. Unit Pressures on a Vertical Plane 
at Depth x below a uniformly Sloping Sur¬ 
face, the Direction of the Pressure being 






105 


taken Parallel to the Line of Greatest De¬ 
clivity. — As in Art. 35 we shall consider a 
wedge of thrust of infinitesimal dimensions, 
of which the left face AB (Fig. 10) is 
vertical and equal to unity, and the upper 
surface parallel to the top slope. The 
weight of the vertical prism that rests upon 
any trial base as BC is, e. BC. cos i. x. == 
e. AT. BC. x (Fig. 9) ; so that neglecting 
the weight of the infinitely small wedge ABC 
we get the value of E from equation (1) of 
Art. 37 by simply replacing G by this value. 
Equation (2) of Art. 38 is thus replaced by 

E = ex .AT. BC — 

AI 

which is exactly that given in Art. 38 multi¬ 
plied by the constant 2x. All the subsequent 
reductions, therefore, hold if we simply put 
h = l = 1 in the final equations, and multi¬ 
ply the result by 2x. Hence in (19) above, 

on changing the co-efficient -li 2 to ex , we 

O O b) 

have the intensity of the pressure at a 
depth x; and on integrating this expression, 
multiplied by c lx between the limits o and h, 



10G 


we are at once conducted to (19), which is 
thus proved true by the method of integra¬ 
tion of the effects of earth particles, which 
is independent of the assumption of a plane 
surface of rupture extending to the surface. 

Precisely the same conclusions hold for a 
vertical wall, or one leaning forwards, when 
E is assumed to make the angle f' or <f> with 
the normal to the wcdl, since G is simply 
replaced as before by the weight of the 
vertical prism for a uniform top slope, and 
ultimately we replace h? by 2x in the general 
formula (11) to get the intensity of pressure 
in the direction given, at the depth x from 
the surface, so that on integrating as before 
we deduce (11) without the necessity of 
considering the surface of rupture as 
extending to the surface. The graphical 
method, using this hypothesis, should again 
give good results. It is possible though, in 
this case, that the influence of the wall 
friction may have some effect in deflecting 
the weights of the vertical prisms from a 
vertical line ; for, when it is so transmitted, 
the usual direction of the pressure is parallel 
to the surface (Art. 9). For wcdls leaning 


107 


backivctrds the prisms do not rest vertically 
over the bases of the prisms of thrust, and 
the theory would seem to be inapplicable ; 
so that the formulae for this ease, (8) and 
(9), have to rest upon the unproved 
hypothesis of a plane surface of rupture 
extending to the surface, and may depart 
considerably from the truth. We conclude, 
that, except for the cases for which 
Rankine’s formula is applicable, the plane 
surface of rupture is still an unproved 
hypothesis. 

49. Point of Application of the Thrust; 
Uniform Sloj)e. —We have the thrust by 
(11), whether the wall inclines forward or 
backward or is vertical, expressed by the 
relation, 

E = Keh\ 

whence the thrust on the area dli x 1 is 
nearly 

dE = 2Kehdh , 

and the vertical distance from where the 
inner face of the wall intersects the top 
slope to the centre of pressure is equal to 
the limit of the sum of the elementary 


108 


pressures multiplied by their arms, divided 
by the total pressure, or, 


h 



hence the centre of pressure on the wall is 
±h vertically above the base. 

50. Surcharge uniformly distributed. — 
If the filling of height li has a horizontal 
surface upon which a uniform load of any 
kind rests, replace its weight by that of an 
equivalent quantity of earth, giving the 
total load the same, and call the height of 
the reduced load li. The total pressure on 
the vertical wall of height li is now 
E = lie ({h + li) 2 - It ' 2 ) = Kell (h + 2 h')> 
whence, 

dE = He.2 (It + li) dli; 

and the distance of the centre of pressure 
from the top of the wall downwards is, 


2 J (h + H) Mh 


h / 2 h + 3 Ji\ . 
3 V h + 21 i ) ; 


o 


li (h -f- 2/i') 


or from the base of the wall up wards, 





109 


+ * kk ' =(i + .*: \*. 

3 h + 6/i' \ /a + 2/i7 3 

It is more than probable that the theory 
for this case will prove illusory in practice, 
and will give a large excess of pressure ; 
so that, most frequently, such surcharged 
loads are ordinarily allowed for by a large 
factor of safety, particularly where the earth 
is bound by cross-ties, stringers, etc., or 
the surcharge is not free to move laterally 
as well as vertically. 

51. Moments of the Thrust about the 
Inner Toe of the Wall. — Let us decompose 
the thrust E against the wall into two com¬ 
ponents, E x and E 2 , respectively normal to 
and acting along the inner face of the wall. 
If E makes the angle (/>' with the normal 
to the wall, we have, from, E = Keli 2 , 

E 1 — E cos (f>' = K cos (p'.eJr ; 

or putting, K x = K cos </>' 

we have, E 1 = K x eh 2 ; 

also, E 2 = E sin <£' = E 1 tan </>'. 

It is understood in these formulae, that, 
when <// > </>, we must replace c/>' by </>. 




110 


If the inner face of the wall makes an 
angle a with the vertical, we have the thrust 
acting at a distance cl = ch sec a from the 
inner toe of the wall, where c = ^ by theory 
for a uniform slope ; therefore, the moment 
M of the thrust, about the inner toe of the 
wall is E 1 cl, since the moment of E 2 is 
zero ; or putting for abbreviation, 

m = c K x sec a 

we have, 

M — E 1 ch sec a = c K x sec a . eh 3 = meh 8 . 

The two formulas for the normal compo¬ 
nent of the thrust and moment, E x = K x > h 2 
and M = meh 3 will be used throughout the 
next chapter, as semi-empirical formulae, 
in which the constants c, K x , and m are to 
be determined by experiment; and it is w> .* 
to recall that h represents the vertical heigh.t 
from the inner toe of the wall to where the 
line of the inner face pierces the top sur¬ 
face of the earth backing, and that e repre¬ 
sents the weight per cubic foot of earth. 



CHAPTER IV. 


EXPERIMENTAL METHODS. — COMPARISON 
WITH THEORY. 

The Practical Designing of Retaining- Walls. 

52. Experiments upon earth-pressure 
have been made by a number of engineers 
and others at various times, but none have 
approached in completeness those made by 
M. M. L. Leygue, and described in 
“ Annales des Fonts et Chaussees ” for 
November, 1885. From them, in fact, can 
be deduced, for the materials used, a com¬ 
plete method for the practical designing of 
retaining-walls, as will be more fully appre¬ 
ciated as we proceed. We only have space 
to indicate the more important results, 
which are given briefly below. 

53. Cohesion. — Leygue first proved that 
cohesion in a mass of sand does exist, but 
that for this material it can often be neg- 


m 


112 


lected in comparison with friction. Thus for 
a vertical mass about a foot high, and ten 
square feet base, the ratio of the resistance 



of cohesion to friction, to sliding on the 
base, is only one-tenth. 

54. The Prism of Rupture. — To observe 
the movement of the sand at the moment of 
























113 


sliding or overturning of the retaining-wall,. 
a wooden box was used (Fig. 11), 1.31 feet 
long, 2.62 feet wide, and 2.4 feet high, 
open above and in front, but with glass sides. 
Movable boards in front, pivoted at the 
bottom, represent the retaining-walls. They 
were 1.22 to 1.3 feet long, and 0.66 to 0.82 
feet high, and were first put at the inclina¬ 
tion desired, when the sand (whose natural 
slope was one and a half base to one rise) 
was placed in the box behind them. Ta 
note the relative movement of the grains of 
sand, as the wall moves over, thin horizon¬ 
tal strata of white plaster pulverized were 
interposed in the mass of gray sand at vari¬ 
ous heights. 

In all the experiments, narrow strips of 
paper, bent at a right angle, were placed 
over the joints to keep out grains of sand, 
and allow of free overturning or sliding. 

As the retaining-board was moved over, 
the following facts were observed : — 

All of the particles lying between the wall 
and a convex surface (shown by a dotted 
line in the figure) moved parallel to this 
surface , which may be called the surface of 


rupture. This curve was determined by 
noting where the thin layers of plaster be¬ 
gan to break, which was easily done; it 
remained invariable as the rotation of the 
wall was increased, even lip to 30° for a 
horizontal surface of earth ; but on com¬ 
pleting the rotation by lowering the retain- 
ing-board to the horizontal, the curve moved 
slightly to the rear, and the earth finally 
took its natural slope. In this first move¬ 
ment, the particles, moving all parallel to 
the surface of rupture, had velocities pro¬ 
portional to their distances from this surface. 
In fact, since the mass to the left of the 
curved surface was invariable during a part 
of the rotation, it follows that the areas of 
the triangular bases remained constant, and 
that their vertices (the intersection of the 
surface of earth with the wall) must move 
parallel to the curved surface of rupture. 
The full friction of the earth teas exerted 
against the waif so much so, in fact, as to 
raise the earth next the wall slightly, and 
thereby diminish its density. 

The convex surface of rupture, roughly 
speaking, bisects the distance from where 



115 


the upper surface meets the wall, to where 
the horizontal through the last point 
meets the line of natural slope through the 
foot of the wall; thus for high surcharges 
differing essentially from theory. 

The same conclusions were found to hold 
when the retaining-wall was moved parallel 
to itself, except that the curve of disjunc¬ 
tion tended from the start to move towards 
the interior of the mass. 

Similar conclusions were found to hold 
for millet seed or for large rocks as back¬ 
ing ; and the surfaces of the bases of the 
prisms of rupture were found to be similar 
figures for heights varying from 0.66 to 
6.56 feet, so that their areas can be sup¬ 
posed to vary directly as the squares of 
their heights in accordance with theory. 

The angle of friction of earth on the 
rough wooden board was found to be 
(fY = 39°, for earth on glass (f / = 24° 30'; 
and yet, when a glass wall was used, the 
surfaces of the mass of rupture were in¬ 
creased only about six per cent, or they 
were practically the same. 

These surfaces for millet seed, whose 


116 


natural slope was one-half base to one rise, 
were thirty-seven per cent greater than for 
sand with a horizontal top surface, and about 
double for a surcharge sloping at one-half to 
one. The following table gives the values 
of x and y (Fig. 11) for h = 1 for different 
inclinations of the surcharge i to the hori¬ 
zontal, and inclinations a of the wall to the 
vertical, counted positive where the wall is 
to the right of the vertical or leans towards 
the earth, and negative when to the left; 
also the greatest departure / of the curve 
of disjunction from its chord, and the areas 
of the bases of the prisms of rupture, all 
referring to sand whose angle, of repose was 
<p = 33°42 / , or a slope of 1^ to 1 : — 


117 


Table No. 1. 


r 











Areas of Bases 

tan f 

tan a 

V 

X 

f 

of Prisms 






of Rupture. 

0 

_ 1 

3 

0 

0.552 

0.088 

0.394 


0 

0 

0.662 

0.053 

.290 


+ :J 

0 

0.786 

0.031 

.201 


1 

2 

0 

0.858 

0.022 

.161 


<2 

'3 

0 

0.933 

0.015 

.127 


_i 

3 

0.885 

1.437 

0.120 

0.853 


0 

0.662 

1.324 

0 077 

.558 


+ * 

0.453 

1.239 

0.047 

.322 


1 

•2 

0.358 

1.216 

0.033 

.236 


2 

S 

0.267 

1.200 

0. 

.172 

f 

_ t 

IT 

1.770 

2.322 

0.150 

1.338 

0 

1.324 

1.986 

0.096 

.818 


+ i 

1.161 

1.692 

0.059 

.454 


l 

-2 

0.906 

1.574 

0.040 

.319 


2 

IT 

0.716 

1.437 

0. 

.220 

• 

ini n 

_J 

1 V 


hu 

• 

For any 

vain, of XiST? 

bav Stop o 

1, we find the cor- 


responding values of a*, y, and f by multiply¬ 
ing the tabular values by 3, and for the areas 
by multiplying by 3 2 = 9. Intermediate 
values are easily obtained by interpolation. 

55. Surcharge uniformly distributed. — 
If we conceive a vertical wall backed by 
earth level with its top, of height /*, with a 
further surcharge of earth vertically over 
the first of a uniform height h', and pre¬ 
vented from spreading by the sides of the 
















118 


box, and a movable board in front in 
prolongation of the first retaining-wall, we 
have the conditions of the experiment. As 
the lower wall was rotated, the earth 
escaped from under the other, and the sur¬ 
face of rupture, which still passed through 
the foot of the lower wall, was no longer 
found to be a simple convex curve. The 
lower part was as before ; but near the top 
of the lower wall it deflected nearly verti¬ 
cally upwards for a short distance, and 
then again towards the interior, thus pre¬ 
senting two points of inflection, and giving 
a much less surface of rupture than theory 
would seem to demand. It is true, though, 
that much of the weight of the surcharge 
was held by friction by the upper wall, so 
that theory no longer applies. (See experi¬ 
ments recorded in “Engineering News’* 
for May 15 and 29, 1886, giving, by weigh¬ 
ing, the exact weight of sand held up by 
the sides of a box when the bottom was 
gradually lowered.) 

56. Values of the Moment m as obtained 
by Experiment. — To ascertain the moment 
of the earth-thrust against the foot of the 


119 


inner face of the retaining-board (Fig. 11), 
the force applied at the upper end of the 
frame constituting the extension of the 
board, and perpendicular to its direction, 
that was necessary for exact equilibrium, 
as overturning was just about to begin, was 
measured by the tension of a rope passing 
over one or two pulleys, and weighted at 
the further end by a weight F. The weight 
of the board and frame, when inclined,* was 
balanced by the pull of a rope at top. 
Allowing for the friction of the pulleys, as 
found by experiment the resisting force 
applied at the top of the frame, 2.46 feet 
from the toe of the wall, was 1.05 F on an 
average, and its moment about the toe was 
1.05 F X 2.46. This must balance the 
moment of the earth-thrust, which, for a 
length pressed of 1.3 feet, would be, if . 
the sides of the box did not cause friction, 
meh s x 1.3 (Art. 51), whence m = 1.284 F, 
on substituting the numerical values given. 
But the lateral pressure of the sand against 
the sides of the box causes friction, and 
thus hinders the forward motion of the sand, 
and thus diminishes the thrust. To estimate 


120 


its influence, Leygue follows Darwin’s plan 
of measuring the force F' (corresponding 
to F) when a partition-board of thickness t 
is placed perpendicular to the wall, and 
centrally in the mass. As the retaining- 
board does not receive the full pressure on 
its length, 1.3 feet, which call d, we can 
suppose, for no central board, the thrust 
exerted on a less length Ad = (d — 2x) to 
correspond to a full thrust if the sides 
exerted no influence. Here A is a co-efficient 
of contraction analogous to that used in the 
case of liquids, and x represents the total 
contraction caused by one side of the box. 
The ratio of the forces F and F' will, 
therefore, be, 

F _ d — 2x 

F 1 d — t — 4a;’ 

since the full thrust, when the centre board 
is used, acts only on the length (d — t — 4a;) 
as the influence of the friction of four side 
walls has now to be considered. The forces 
F and F' were directly measured, and the 
values of x found from the last equation, 
which, substituted in the preceding equation. 



121 


gave the co-efficients of contraction A. For 
level-topped earth A was found to amount 
to 5%, and for the surface sloping at the 
angle of repose to as much as 15%, which 
illustrates the marked influence of counter¬ 
forts in diminishing the actual thrust exerted 
against the face wall. The results can be 
represented by the formula, 

A = 0.135 (7— tan i) ;• 

hence the true value of m, corresponding to 
the full thrust if there were no side walls, 
is to be found from the equation, 

1.05 F x 2.46 

= 1.3 x meh s X 0.135 (7 — tan i). 

57. The values of F , as found by experi¬ 
ment, corresponding to h = 0.66 feet or 
less, and e — 89 pounds per cubic foot, 
enable us to compute the values of m given 
in the next table. 

The values of m for millet-seed, whose 
angle of repose is 26° 34', or tan <£ = 
are likewise inserted, being obtained by 
experiment in the way indicated above. 


122 


Table ISTo. 2. 


B 

Values 

of m for Sand. 

Values of m for 
Millet-Seed. 

5 

*-* 

tan / = 0 . 

tan i = 

tan i = |. 

tan i = 0 . 

tan i = 5 

-I 

0.535 

0.970 

1.329 



_i 

T 

.241 

.425 

.574 

— 

— 

_2 

.152 

.260 

.356 

- 

— 


.063 

.103 

.136 

0.084 

0.154 

0 

.030 

.047 

.065 

.057 

.106 

i 

.015 

.024 

.032 

.037 

.063 

.009 

.014 

.018 

— 

— 

i 

0 

0 

0 




58. To these most valuable observations 
are to be added those of Mr. George 
Darwin, Trinity College, Cambridge, made 
in a box 0.72 feet long, 1.8 feet high, and 1 
foot wide, after the manner explained above* 

Vertical wall , backing in horizon¬ 
tal layers not compacted . . . M = 0.030 eh 3 - 

Do. compacted filling. M = .022 eh 3 

Vertical wall, backing sloping back¬ 
wards at 3 to 2. . : . . . . M = .0275 eh 3 

Do. layers sloping upwards at 3 to 2, M = .0315e/t a 

Mean for loose earth. M — 0.030 eli 3 

The co-efficient 0.030 is the same as that 


















123 



given above for level-topped earth ; but for a 
surcharge sloping at the angle of repose 33° 
42', Mr. Darwin found m = 0.048, whereas 
Leygue’s constant = 0.065. Leygue sug¬ 
gests that this difference is to be accounted 
for from the small size of the box, which 
caused the prism of thrust to be truncated- 
It is doubtful whether Leygue’s constant 
is great enough for a mass sloping indefi¬ 
nitely, for the prism of rupture for earth at 
the natural slope was barely contained in. 
his box. Still, theory does not indicate a 
marked increase in the thrust for very high 
surcharges, over those about equal to the 
height of wall; but we have seen in Art. 41 
that theory gives, for the surface sloping at 
the angle of repose, 33° 42', a thrust nearly 
double that corresponding to a surface 
slope of only 7° less in place of the ratio 
65 to 47 given in the table. It is just here 
that we should expect, and will find, the 
greatest divergencies between theory and 
experiment; for cohesion is much more 
marked for a long surface of rupture than 
for a short one, which doubtless causes the 
surface of rupture in the experiments to 


lie much nearer the vertical than theory 
would indicate. This divergence becomes 
still more marked for walls leaning back¬ 
wards, as we shall see farther on. 

59. The experiments of M. Gobin (1883) 
were of the same character as those of 
Darwin and Leygue, though not so exten¬ 
sive as those of the latter. Leygue’s re¬ 
sults are about 14% inferior to Gobiu’s, 
which Leygue asserts is due to the stiffness 
of the cards used, and the large (20% of 
the load) friction resistance of the pulleys, 
the stiffness of the hinges used at the foot 
of the board, manner of filling in, and 
lastly the appreciation of the instant of 
sensible movement. 

60. Direct Estimation of the Normal 
Component of the Earth-Thrust, and of its 
Point of Application. — To estimate the 
normal component of the thrust, the same 
box (Fig. 11) was used; but the single 
board was replaced by a frame containing 
two parallel boards separated by springs, 
whose compression gave a direct measure 
of the thrust. The normal component of 
the weight of the inner movable panel was 


125 


equilibrated by the pull of a rope passing 
over a pulley when the panel was inclined 
to the rear, and directly measured by the 
springs, and allowed for when it was inclined 
forward. The movable panel was likewise- 
equilibrated for the friction caused by the 
rubbing of the earth in its double move- 
ment of sinking and translation, by the pull 
of ropes acting upwards in the plane of the 
panel, which thus prevented the scraping 
of the panel on the bottom of the box. 
Direct experiment showed that the force 
necessary to slide the panel upwards, when 
the full earth-thrust was exerted against it, 
was about equal to the normal component E r 
multiplied by the co-efficient of friction 
/ = tan <£, as has been hitherto assumed; 
so that, to get the resultant pressure on any 
wall, we must combine the value of E t 
found by aid of the following table with 
E x tan cf), which thus gives the direction of 
the pressure as inclined below the normal 
to the wall at the angle <£. The method of 
experimenting was to fill in behind the 
movable panel, equilibrated as explained, 
and kept temporarily at the same distance 


126 


from the fixed panel by a fastening; then 
the fastening was suddenly removed, and 
the earth suddenly exerted its thrust, so 
that the compression of the springs follow¬ 
ing was double that due to the constant 
thrust of the earth. Calling half the force 
measured by the springs E l on a length d, 
and for the vertical projection of the area 
pressed equal to h x d, we have, admitting 
the law of the parabola (Art. 51), 

E l = K x e/i 2 Ad, 

where A, as before, is a co-efficient of con¬ 
traction. The value of A was determined, 
as explained in Art. 57, by experiments 
with the spring apparatus, to be, A = 0.120 
(7 — tan 1). 

From these formuhe the value of K x was 
obtained as given in the following table. 
By multiplying K x by sec </>, we obtain the 
co-efficient K in the formula for the total 
earth-thrust, E = Kelt 2 , on a wall one foot 
long. 

By taking the foot of the movable panel 
as a centre of moments, we have the 
moments of the forces recorded by the 


127 


springs equal to the moment of the double 
earth-thrust, from which the point of appli¬ 
cation of the latter can be readily found. 
Calling the fraction of the height from the 
base to the point of application of the earth- 
thrust c, we thus obtain the quantities in the 
column headed c in the next table. 


Table Xo. 


3. 


tan a 

Spring Appa¬ 
ratus. 

c 

Spring Appa¬ 
ratus. 

A', 

K x a 

tan i= 0 

1 

a 

.*2 

3 

tan *=0 

1 

5 

9 

5 

tan i— 0 

1 

1 

o 

3 

_ 3 

5 

0.418 

0.461 

0.485 

0.712 

1.223 

1.519 




_ 1 

1 

.432 

.467 

.486 

.390 

.645 

.836 




o 

5 

.439 

.467 

.485 

.243 

.367 

.508 




_ 1 

3 

.437 

.462 

.480 

.136 

.211 

.271 

0.165 

0.271 

0.366 

0 

.425 

.44S 

.470 

.071 

.104 

.137 

.094 

.147 

.188 

1 

5 

.401 

.429 

.455 

.038 

.054 

.067 

.048 

.071 

.090 

2 

t 

.383 

.411 

.438 

.019 

.029 

.034 

.023 

.032 

.040 


61. It seemed to the writer that it would 
be of interest and value to use the areas of 
the bases of the prisms of rupture (Art. 54) 
to represent O in Pbg. 3, Art. 3, and by 
the method indicated there to find E and 
E 1 = K* when e = 1 and h = 1. 


f 























128 


The actual surfaces given in Leygue’s 
tables were used, and the constructions 
effected on his drawings, using the chord 
of the line of rupture as giving the average 
direction of the corresponding surfaces of 
rupture. 

The values thus found of this normal 
component, which is termed 7T 1 a , are given 
in the table just to the right of the values 
of K x as given by the spring apparatus for 
•comparison, and it will be observed that 
they are uniformly larger than the values of 
K v To find the distance c tt to the point 
•of application of the thrust K* for h = 1, 
we have only to divide m of the table of 
Art. 57, by K*, and the results are given 
in the following table. As it is of value in 
this connection, the same thing was done 
for millet-seed, where tan = J, cj) = 26° 
34'; the moments being taken from the table, 
Art. 57, and the thrusts K* being found by 
taking the volumes of the prisms of rupture 
as given by experiment, and finding E x from 
the construction shown in Fig 3. 


129 


Table No. 4. 


tan a 

tan i 

For Sand. 

c «- 

AY' 

Millet-Seed. 
r a — m 

A\ a 

By 

Theory. 

c 

_ l 

3 

• 0 

0.382 

0.402 

i 

3 

1 

3 

] 

2 

.3*0 

.383 


i 

3 

2 

15 

.371 


i 

0 

0 

.319 

.420 

l 

3 

0 

l 

V 

.326 

.450 

i 

j 

0 

2 

3 

.346 


i 

3 

1 

3' 

0 

.312 

.410 

1 

3 

i 

3 

1 

2 

.338 

.450 

1 

3 

l 

15 

1 

• 3oo 


i 

3 

Q 

3 

0 

.391 


l 

3 

2 

15 

1 

2 

.440 


3 

1 

2 

a 

.450 


3 

Average. . 

0.367 

0.419 



Since the actual wedges of rupture, by 
experiment, vary as the squares of the 
heights, c a should equal one third, from 
which it does not differ very greatly for sand. 
We should conclude from the value of c given 
by the apparatus with springs, that the 
point of application of the thrust was con¬ 
siderably above the li given by theory, 
lying generally between 0.4 and 0.5, which 
last value it can never equal. But, taking 

















130 


the thrusts clue to the actual wedges of 
rupture, the values are much nearer the 
theoretical. Neither method is exact, since 
in the first case, as the sand advanced, it 
rubbed on the bottom, and thus hindered 
the advance there more than at the top ; 
and, in the second case, the wedge of rupture 
cannot act as an invariable solid sliding 
down two planes at once as theory requires. 1 

It can be said, though, in support of 
Leygue’s values of K x and c, that their 
product gives almost exactly the value of 
m (Art. ;37), determined in an entirely dif¬ 
ferent manner, and the results are of course 
correct for a wuxll that has slid about the 
average compression of the springs; but it 


1 Perhaps a correct value of the horizontal thrust could 
be obtained by putting a wide, though light, retaining-wall 
on wheels, and attaching cords to the bottom horizontally to 
the rear, where they could pass over pulleys, and weights be 
attached sufficient to prevent motion; then, after filling in 
the sand behind the wall, the weights can be diminished 
until motion just begins, and the horizontal component of 
the thrust read off, allowing for friction. The wheels could 
preferably be of the kind used on sliding doors to diminish 
friction as much as possible, as the vertical component of 
the thrust would add to it considerably, which, however, 
could be equilibrated as in Leygue’s experiments. 





131 


is possible that the initial thrust before the 
wall gives is somewhat different. 

The average of c given by Leygue’s 
spring apparatus for millet-seed was 0.417, 
or less than for sand ; and he therefore con¬ 
cludes that c approaches its theoretical value 
as cf) diminishes. 

62. Influence of the Character of the 
Retaining-Wall. —When the retaining- 
boards were of glass, it was found that, on 
an average, 



b= 1.02 and — = 1.053, 


m 


where c', AY, and m' refer to the quantities 
obtained with a glass wall, and c, K v and m 
to the roughened wooden one. For the 
wood on sand, tan <// = 0.810, and for the 
glass, tan cf)' = 0.45G. 

63. Surcharge uniformly distributed. — 
For the case of Art. 55, the following 
■values were found with the apparatuses 
before used : — 



132 


Table No. 5. 


Height of 
Surcharge. 

h' 

Centre of 
Pressure. 

c 8 

AY 

m s 

0.00 

0.425 

0.070 

0.030 

0.5 h 

0.434 

0.094 

0.042 

1 x h 

0.440 

0.117 

0.051 


These are much lower values than the ordi¬ 
nary theory of Art. 50 would give, which r 
however, is hardly applicable here. 

As h' varies between 0 and /<, we can 
express the values of c, K v and m by the 
following empirical formulae : — 

c 8 = c(l + 0.035 J £j ; K* = 

1 + 0.60 ; m* = m 

where c, K v and m refer to It — 0. 

For h = //, the value of E given by the 
formula of Art. 50 is about double that found 
by experiment, and the centre of pressure 
lies above in the ratio of 1.333 to 1.035. 

64. Comparison with Theory. — In the 


( 


,+0 'i 












133 


following table are given the theoretical 
values, side by side with those obtained 
from actual experiment, for the angle that 
the plane of rupture makes with the hori¬ 
zontal, and also the values of K and m in 
the formulas for the total thrust, E — E x 
sec cf> = Keh 2 = K x sec cj> . eh 2 (Arts. 42 
and 51), and for the moment about the 
inner toe of the wall, M = meli 3 . Tan a 
is regarded as minus when the wall is in¬ 
clined away from the earth, and plus when 
inclined towards the earth. 


Table No. 6. 


tan a 

tan i 

Angle of Rup¬ 
ture WITH THE 
Horizontal. 

Coefficient 
A” OF 

the Thrust. 

E—Keh % . 

Coefficient 
m of 

overturning 

Moment. 

M = meh 3 . 

Theory. 

Experi¬ 

ment. 

Theory 

Experi¬ 

ment. 

Theory. 

Experi 

ment. 

_i 

3 

0 

60° 2T 

61° 10' 

0.222 

0.163 

0.065 

0.063 


1 

2 

50° 

58° 10' 

.411 

.254 

.120 

.103 


2 

3 

33° 40' 

57° 

.750 

.325 

.232 

.136 

0 

0 

56° 36' 

56° 30' 

0.133 

0.085 

0.037 

0.030 


i 

47° 30' 

54° 50' 

.214 

.125 

.060 

.047 


2 

3 

33° 40' 

54° 10' 

.416 

.165 

.115 

.065 

1 

3 

0 

50° 12' 

51° 50' 

0.072 

0.046 

0.021 

0.015 


i 

3 

44° 48' 

51° 10' 

.109 

.065 

.032 

.024 


<2 

3 

33° 40' 

50° 50' 

.218 

.082 

.064 

.032 




















134 


It will be observed that for i = 0, the 
values of the angles of rupture as given by 
experiment (average inclination of the 
curved surface of rupture was taken) agree 
well with theory, as also the values of m ; 
but for surcharged walls the theoretical and 
actual values differ materially, especially for 
tan i = §, for which value for a leaning 
wall the theoretical thrust is double the 
actual. This has been partly foreseen, since 
for tan i = f, the theoretical plane of rup¬ 
ture approaches indefinitely the natural slope 
in direction, which calls for an infinitely 
great wedge of rupture. The theoretical 
thrust, however, does not differ materially 
from one corresponding to a plane of rup¬ 
ture lying much nearer the vertical, as has 
been hitherto mentioned ; so that the differ¬ 
ence must be due to some other cause, in 
which cohesion, of course, plays some part. 

Whatever may be thought of the values 
of c and K x obtained by the spring appa¬ 
ratus, there can be no question about the 
accuracy of the value of m, which is of 
predominating influence where overturning 
is considered. 


135 


65. Stability of Retaining- Walls against 
Overturning. — To illustrate the use of 
Leygue’s constants in testing the stability 
of a retaining-wall of height li (Art. 51) by 
a graphical method, we first compute the 
values E = E 1 sec 4>' = sec (p' eh 2 (put 
(f> for cfV when cf>' > (f>) and ch from the 
assumed values e and h, and the values of 
the constants c and K x derived from the 
table (Art. 60) ; then lay off to scale E 
from the inner face of the wall, at a vertical 
height equal to ch above the base, and 
making an angle below the normal to the 
inner face equal to ft (or <£ if <f>' > 4>) as in 
Fig. 3, and then combine this thrust with 
the weight of the wall W, acting along the 
vertical through its centre of gravity, to 
find the resultant on the base. Its inter¬ 
section with the base gives the centre of 
pressure on the base ; and this point should 
at least lie within the middle third of the 
base, for reasons given in Chap. I. The 
wall need not be drawn for more than half 
its height; and, if the drawing is made to 
a large scale, the results are very accurate. 
This graphical method should always bo- 


136 


used in testing the results as given by the 
method of moments for stability against 
overturning. 

To illustrate the use of the method of 
moments , let us consider a wall leaning 
towards the earth, the back making the 
angle a with the vertical, and suppose the 
base perpendicular to the back, and of 
thickness t. 

We may consider three forces as acting 
on the wall: the normal component E v 
acting at a height cli above the base, or at 
a perpendicular distance ch sec a from the 
outer toe ; the friction E x tan <f>' (use </> for 
<f>' when (fV > (jy) of the earth on the wall, 
acting downwards along the back of the 
wall at a distance t from the outer toe ; and, 
lastly, the weight W of the wall, acting 
along the vertical through its centre of 
gravity, and at a horizontal distance g from 
the outer toe. 

If we call <7 the co-efficient of stability, 
or the factor by which it is necessary to 
multiply the normal component (leaving the 
friction at the back of the wall the same) 
to cause the resultant on the base to pass 


137 


through the outer toe, we have, taking 
moments about the outer toe, 

Wg — E 1 (cr ch sec a — tan <£'.£) 

= K x a c sec a eh s — E 2 t 

. •. Wg = cr meh 3 — E. 2 t ; 

since (Art. 51), c sec a 

= m and E x tan <$>' = E v 

The factor of safety, o-, was not made to 
increase the total friction at the back of 
the wall, because the usual causes which 
influence and increase the thrust are passing 
loads and rains ; and the latter, in lubri¬ 
cating the surfaces of contact, would prob¬ 
ably cause the total thrust on the wall to 
approach nearer the normal than before, 
so that is not safe to consider the friction 
increased in the same ratio as the normal 
thrust. 

The formula skives m as the controlling 
constant in the right member. If we take 
this constant from the table in Art. 57, 
where it was accurately found from experi¬ 
ments on rotating retaining-boards, the 
result should closely agree with the reality; 


138 


for the term E z t is of minor importance, 
and in case Leygue’s values of E v and 
•consequently of E 2 — E 1 tan <£', are too 
small, the only effect is to add to the 
stability of the wall designed by the above 
formula, and is therefore on the side of 
safety. 

From the above formula we can ascertain 
the factor of safety a for a given wall, or 
for a given stability find the thickness t 
(on expressing W and g as a function of £), 
when the other quantities are assumed. 
For cr = 1 we have exact equilibrium 
about the outer toe, supposing no crushing 
there. 

For a given design to find the distance u 
from the inner toe of the wall to the centre 
of pressure on the base, call g' = horizontal 
distance from the inner toe to the vertical 
through the centre of gravity of the wall, 
and we have the equality of moments, about 
the centre of pressure, of IF, E v and E 2 
expressed as follows : —* 

IF (u cos a — g') — meh 3 — u E z , 
from which u can be found, and the centre 


139 


of pressure located. If u = § t, there will 
be no stress at the Inner toe (Art. 17), and 
for greater values of u the joint would tend 
to open. The wall can be designed by this 
formula for any given value of u expressed 
in terms of if if preferred. 

66. Stability of a Retaining - Wall against 
Sliding. — The simple test in this case is, 
that the resultant on the base of the wall 
shall make, with the normal to the base, 
an angle less than the angle of friction 
between the wall and its foundation, — a 
condition that can always be satisfied by 
sufficiently tilting up the base in front, 
which safeguard should never be neglected, 
particularly for dock or river walls, which 
generally fail, if at all, by sliding. The 
angle of friction of masonry on wet clay is 
only about 18°. 

It is equally necessary to go down deep 
enough for a firm foundation, or to drive 
piles, preferably in the direction of the 
resultant on the base. The foundation- 
course is generally made larger than the base 
of the wall, to better distribute the pressure 
on the base. 


140 


67. Discussion of Experiments on Re¬ 
taining-Walls. — In the following table 
is given a resume of quantities relating to 
a number of experimental retaining-walls 
all at the limit of stability by actual experi¬ 
ment. 

The earth-thrust against the various 
walls has been computed by three different 
methods : — 

1st, By theory (Chaps. II. and III.). 

2d, By using the thrust resulting from 
the actual wedge of rupture for sand as 
given in Art. 61, using value Kf. In 
both these cases the point of application 
was taken at one-third height of wall from 
the base. 

3d, By Leygue’s method, using the con¬ 
stants 7ij and c for sand (c/> = 33° 42'), 
given in Art. 60. 

From the insufficiency of our experi¬ 
mental data for all materials, varying in 
their angle of repose within practical limits, 
the same constants had to be used, by the 
second and third methods, for varying 
values of (when between 33° and 40°), 
which strictly apply only to </> = 33° 42'. 


141 



Table No. 


















































142 


For brevity we denote by, — 
h the height of the wall in feet, 
t its thickness at the base, 
e the weight per cubic foot of earth, 
w the weight per cubic foot of wall, 
a the angle the inner face of the wall makes 

with the vertical counted j j according 

as the wall leans \ towart ^ s i the earth, 

( from ) 

t the angle with the horizontal of the top sur¬ 
face of the backing, 

0 the angle of repose of the backing, 

<p' the angle of friction of earth on wall, 
q the ratio of the distance from the centre of 
pressure on the base of the wall to the outer 
toe, to the thickness t, using first method 
above (theory), 

q' ditto, using second method, or actual wedge 
of rupture (for sand) method', 
q" ditto, using Leygue’s, or the third method. 

The walls were all of uniform cross-section, 
except Nos. 8, 9, and 10. Walls 1, 3,4, 8, 9,10, were 
of wood with sand backing, except the first which 
had shingle backing. Lieut. Hope’s wall, No. 2, was 
of bricks, with ordinary earth for backing. Curie’s 
wall, No. 7, was in Portland, cement backed by 
damp sand; but Nos. 8, 9, and 10 were peculiar 
triangular frames, whose inner faces made the 
angles 27° 30' and 55° with the vertical (see 
Curie’s “Pouss^e des Terres,” Paris, 1870, and. 


143 


“Troisnotes” in 1873). In No. 8 the face exactly 
coincided with the “limiting plane” (Arts. 28 and 
41), and in Nos. 9 and 10 was below it; so that the 
thrust for No. 8 was found as usual, the values of 
■c, K i, and K\ being interpolated from the tables. 
But for Nos. 9 and 10 the thrust was first found 
on the limiting planes, respectively 27° 30' and 
28° 15' to the left of the vertical, which was then 
combined with the weight of earth and frame to 
the left, assuming the thrust on the limiting plane 
to make an angle <i> with the normal, as theory 
demands. 

In wall No. 7 the surcharge extended entirely 
over the top of the wall to a height of 4.26 feet, 
and was then level. The values of q' and q" were 
not found for this wall, since (j> = 45° was so much 
greater than 33° 42', to which q ' and q" refer. In 
wall No. 1 the earth did not reach the top of the 
wall by 0.25 feet. Gen. Pasley’s wall was omitted 
from this list, as possibly the base was imperfect, 
since the theoretical force necessary to apply at 
the top to overturn it was to the actual as 20.6 to 
17.53. The value of q, however, was found to be 
-j- 0.12, and of q' + .08, for h = 2:17 feet, t = 0.67 

feet — = 1.06, a = i — 0°, and $=■-$' = 39°. The 
w 

values of q, q', and q" were found for some stable 
walls, not quite at the limit of stability, to be 
positive as they should be. All of the values of 
h, given in the third column of the table, are 
simply averages of several values found by the 
experimenters under the same conditions. 


144 


It is stated by Flamant, that a retaining-wall 
made of a very light, empty wooden box, of a 
width slightly greater than the height, and 
hindered from sliding by a small obstacle placed 
at the exterior edge, which did not interfere with 
rotation, was just able to support the thrust of 
sand, tilled in to the height of the box. This ex¬ 
periment has afforded ground for ingenious specu¬ 
lation, but is perfectly explained if we take the 
co-efficient of friction of sand on the side of the 
box at about one-third. The experiment should 
have been completed by ascertaining the value of 
this co-efficient. 

An inspection of the average values of 
7 , q' and q" in the table shows that they 
are very near zero, except for walls leaning 
towards the earth, where “ theory ” diverges 
greatly from the reality. For vertical walls , 
with earth level at top , either method gives 
practically the same results when <£ is near 
33° 42', to which the experimental constants 
alone apply. AVe can then use, with con¬ 
fidence, either method. It is likewise seen 

• i 

from experiments 11 to 14 on leaning walls 
that theory fails here, but that q' and q r 
are practically the same ; so that either the 
second or third methods can be used. It 


was an agreeable surprise to find that the 
second method gave as good results as the 
third throughout. As it is a very simple 
thing to find by experiment with any earth, 
the actual wedge of rupture, the practical 
value of this discovery is evident, as it can 
then be used, as explained in the preceding- 
chapters, to evaluate the thrust. 

It is a little disappointing that the third 
method, using Leygue’s constants, does not 
agree more exactly with his experimental 
retaining-walls. To point out the discrep¬ 
ancies more fully, the values of h , for the 
assumed t = 0.1 foot, were computed by 
Leygue’s method (third), and by theory (first 
method), and compared with the average 
value given by his experiments. To judge 
from these experiments on the very small 
model walls, we should naturally infer that 
Leygue’s method gave the thrust too small 
for vertical walls, and too great for walls 
leaning at an inclination of ; but, doubtless, 
these discrepancies will disappear when 
larger walls are used. In fact, for a 
peculiar triangular wall 6.6 feet high, com¬ 
posed of rubble, in a thin mask of wood 


146 


with thin counterforts, his theory applied 
very well; the wall, however, breaking above 
the base until a wooden base was attached, 
when it moved over in one piece. 

Table No.- 8. 


t ASSUMED AND h COMPUTED. 


No. 

Actual. 

t 

By Leygue’s 
Meih..d. 
h 

By 

Experiment 

h 

By Theory. 
h 


Feet. 

Feet. 

Feet. 

Feet. 

5 

0.1 

0.443 

0.307 

0.430 

6 

0.1 

.321 

.285 

.300 

11 

0 l 

.755 

.75* 

.009 

12 

0.1 

.402 

.470 

.371 

13 

0.1 

1.230 

1.310 

.901 

14 

0.1 

.092 

.738 

.450 


Experiments 6 and 7 (see previous table), 
for a surcharged vertical wall, the earth 
sloping at the angle of repose, would seem 
to indicate that theory was verified in spite 
of the fact that the constants by Leygue are 
/ij = 0.137 and m = 0.065, whereas by 
theory 7T, = 0.346, and m = 0.115, for (f> 
= 33 c 42'; but this apparent verification is. 
to an extent accidental, and resulting from 
the special relations of cr and t for these 












147 


cases. If we assume that the masonry 
weighs one-fourth more than the earth, as 
in a following article, then for a = 3 and 
h = 1 we find by the formula of Art. Go 
that t — 0.58 by theory, but only 0.49 by 
Leygue’s method for the surface sloping at 
the angle of repose. For the surface slop¬ 
ing at theory gives t = 0.45, and Leygue 
0.424 ; and for earth level at top, theory 
gives t = 0.367, and Leygue t = 0.343. 

This shows that theory agrees fairly well 
with experiment for gentle slopes, but for 
the surface sloping at the angle of repose 
it gives too large values. It is not a little 
remarkable that with the values of K v and 
above all of m, differing so much from those 
of Leygue, for this case, that the computed 
values of t should agree so closely by the 
two methods. 

Perhaps future experiments, with higher 
surcharges, may cause the difference to dis¬ 
appear entirely. 

The above are undoubtedly the most 
valuable experiments we have, as they refer 
to walls in the ordinary conditions of their 
employment in practice, and should be care¬ 
fully studied. 


148 


G8. General Formula for Stability of 
Retaining - Walls against Overturning. — 
Let Fig. 12 represent a wall ABCD , whose 
length perpendicular to the plane of the 
paper is unity, and whose exterior and 



interior faces and diagonal AC make angles 
with the vertical equal to (3 , a, and re¬ 
spectively. Let W denote the weight of 
the wall, and g the horizontal distance from 
its line of action to the outer toe A; also 
call cr the factor by which it is necessary to 






149 


multiply the normal thrust K^h 2 , leaving 
the friction fK x eli 2 at the back of the wall 
constant, in order that the resultant on the 
base may pass through the outer toe. Here 
f = tan and the quantities /z, t, e, w , i, <f> 
and (f)' have the meanings given in the pre¬ 
ceding article. 

Taking moments around A, we have, 

Wcj -f- fK-fill 2 . t cos a = 

cr K x eh 2 ( ch sec a -f- t sin a). 


We find also, t = li (tan <d — tan a) ; 
and since the moment Wg is the sum of the 
moments of the triangular prism ADI , and 
the rectangular prism IDGE , minus the 
moment of the triangular prism BCE , all of 
the same density zc, we readily find it to 
equal, 


— tan /3 . - h tan /3 -f — (tan 2 w — tan 2 /3) 
2 3 2 


h 2 i “1 

— — tan a It (tan w-tan a) 

2 o J 


w 


or, 


—- (3 tan 2 oj — 3 tan w tan a -f-tan 2 a—tan 2 (3 ). 
fi v 




150 


Observing that m = Kj c sec a, on substi¬ 
tuting these values, and resolving with 
respect to tan w, we find, 
tan 2 at + 


tan 


Cl) 


6 • \ 

2 — Kff COS a — cr sin a) — tan a 


w 


= 2 - 


cr rn + /ij tan a (/ COS a — cr sin a) 

tan 2 a — tan 2 /3 


It will be observed that c does not appear 
in this formula. This formula equally 
applies when the inner face of the wall 
leans away from the earth, or B falls to the 
right of i£, on simply replacing sin a and 
tan a by (— sin a) and (— tan a) through¬ 
out. As this formula is independent of 7/, 
it is true for all values of h. 

G9. The Practical Designing of Retaining- 
Walls. —By the use of various formulas, 
the quantities in the following table were 
made out by Leygue, using the values of m 
and /tj given in Arts. 57 and 00 from his 
experiments, and assuming the values 

— = - and f = tan <£ = 
wo' 3 







151 


The value a in columns 5, 7, and 6 gives 
the ratio of thickness t at base to the height 
h for o- = 1, o- = 3, and the centre of 
pressure on the base at the outer middle 
third limit (column C\) corresponding to no' 
stress at B ; and the lltli column gives the 
corresponding factor of safety for the latter 
case, which, it may be observed, lies between 
1.19 and 3.57, and thus varies between such 
wide limits as to prove that the middle third 
limit is not so good a test of the stability of 
the wall as the use of a constant factor 
of safety <r. 

While, therefore, it is recommended that 
this limit be not exceeded, it is likewise 
recommended that the factor of safety a be 
not taken below say 2.5, which about 
agrees with practice for rectangular walls 
(type 4). 


Table No. 


152 


















































Table No. 9 — Continued. 


153 


b 

r-H 

Co »-H 

2.51 

2.45 

2.40 

1.33 

1.41 

1.46 

1.14 

1.19 

1.23 

Surface S . 

CO 

II s 

b 

0.343 

0.424 

0.490 

1.257 

0.223 

0.189 

0.349 

0.761 

0.155 

0.219 

0.269 

0.643 

o 05 

0.312 

0.376 

0.431 

1.119 

0.121 

0.166 

0.208 

0.495 

0.066 

0.100 

0.129 

0.295 

rH 

II oo 

b 

0.184 

0.224 

0.258 

0.666 

0.093 

0.124 

0.154 

0.371 

0.056 

0.086 

0.106 

0.248 

8 

CO 

II t- 
b 

0.343 

0.424 

0.490 

0.223 

0.289 

0.349 

0.155 

0.219 

0.269 

**|co 

0.312 

0.376 

0.431 

0.121 

0.166 

0.208 

0.066 

0.100 

0.129 

rH 

II 

b 

0.184 
0.224 
0.258 . 

0.093 

0.124 

0.154 

0.056 

0.086 

0.106 

tani 

4 

O «|c<lc<i|oo 

O »h|<nn|co 

® Hc$n|m 

tan a 

3 

o 

+ 

+ 

tan 0 

2 

o 

II 

<n_ 

a 

c3 

b 

II 

< 30 . 

b 

II 

CQ. 

Type 

No. 

1 



tO 
































































154 


The columns 8 , 9, and 10 give the areas 
of the cross sections of the walls for tr = 1 , 
cr = 3, and the middle third limit ( C J) ; and 
it is by an attentive study of these areas 
that the most economical type of wall 
for any case can be found. For cr lying 
between 2 and 3, the types are most 
economical in the following order : — 

Nos. G and 5, leaning walls having parallel 
faces ; 

No. 3, with the exterior face battered, and 
the inner face vertical; 

No. 1, both faces battered ; 

No. 4, both faces vertical; 
and lastly No. 2, with the interior face bat¬ 
tered, and the exterior face vertical. 

Thus the leaning walls are decidedly the 
most economical; and, besides, the press¬ 
ures are better distributed on the base. 
The formula does not apply to type No. 1 
for cr = 1 as /3 > w, but the surfaces for 
o- — 1 were made out for a triangular type, 
having an interior batter of and the angle 
ft as found by computation. 

70. For walls with projections at intervals 



155 


on the exterior or interior, or for buttressed 
or counterforted walls, theory does not show 
any economy over the leaning walls (types 
5 and 6). In fact, for face-walls at the 
limit of stability, except for the buttresses, 
and for a factor of safety of the entire wall 
including buttresses, about a line passing 
through the outer toes of the buttresses of 


Fig. 13. 



from 2 to 3, the surfaces S were about 
equal to those for t} 7 pe 5, corresponding to 
a mean of columns 8 and 10. Fig. 13 
shows a good form of buttressed wall, with 
the face in the form of arches convex from 
the earth side ; but their hideousness militates 
against them. For counterforted walls, the 
lateral pressure of the earth against the 
sides causes friction ; and if (see Art. 56) 













we admit that the thrust on the face-wall is 
diminished y 1 ^, and that the excess is sus¬ 
tained through the friction by the counter¬ 
forts, a computation shows that still the 
counterforted walls are no more economical 
than type 3 with the back vertical and an 
exterior batter. It is likely that the thrust 
is less than assumed, as experiment has 
demonstrated the very great economy of 
counterforted walls. But it is a fact that in 
numerous cases the counterforts have been 
broken off from the face-wall, possibly from 
being carried down, through a bad founda¬ 
tion, by the settling of the earth, as well as 
from the earth-thrust; and they are not to be 
recommended, except as a support to lean¬ 
ing walls, during construction especiall} 7 . 
To push economy to an extreme, a thin, 
leaning face-wall or mask is used, with 
counterforts at intervals, whose back faces 
are plumb; the space between being filled 
with earth well compacted, or with rough 
stones laid by hand. It is found that such 
a wall has nearly the same stability as if the 
compacted earth or rubble was a part of the 
wall (see “ Annales des Fonts et Chaussees 


157 


for November, 1885, and January, 1887, for 
descriptions of such walls constructed in 
Franee). Leygue actually constructed some 
small triangular counterforts in plaster, with 
an exterior slope of without any face- 
wall, which held the earth perfectly. The 
height was 1.31 feet, the base 0.66 feet, and 
the thickness only 0.08 feet; the distances 
between the counterforts being from 0.33 
to 0.49 feet, the intervals between being 
filled with plaster thrown in without especial 
precautions. The earth extended to the top 
of the counterforts, and with a surcharge 
besides. 

In the case of very high walls, it would 
prove economical to connect the counter¬ 
forts at various heights with arches (these 
could easily be constructed with an earth 
centring), which w r ould immobilize a large 
portion of the earth filling, and tend to 
withdraw the active thrust farther to the 
rear. The space between the arches and 
counterforts should be entirely filled with 
compacted earth, to add to the stability. 

71. We have now completed our task of 
giving complete theoretical and semi-empiri- 


\ 


158 


cal methods for the design of retaining- 
walls, and have discussed all available 
experiments by the different methods, and 
compared results, so that the engineer can 
better appreciate their practical value in the 
practical designing of retaining-walls. We 
have found that although theory gave good 
results for vertical walls with earth level at 
the top, yet for many of the other cases it 
departed essentially from practice ; so that 
for all cases the two semi-empirical methods 
are to be recommended as giving: the best 
results, supplemented when necessary for 
various values of (/> by the theoretical 
method. 

From preceding discussions, the engineer 
will doubtless draw some analogy between 
the theory of long columns and that of 
retaining-walls, from their inapplicability in 
certain limiting cases ; and as in the case 
of columns the engineer has come finally 
to depend entirely on experiments, the same 
may be confidently predicted of retaining- 
walls. 

The experiments given in this treatise 
must be regarded, then, as only the first 


159 


instalment on small-sized walls ; and it is 
to be hoped that in the future the experi¬ 
ments will be extended to larger walls, and 
higher surcharges, and that by independent 
methods of observation the amount of the 
earth-thrust, as well as its point of applica¬ 
tion, will be more definitely ascertained. 





APPENDIX. 


DESIGN FOR A VERY HIGH MASONRY 

DAM. 

Engineers are by no means agreed upon 
the proper profile to give high-masonry dams, 
although the three conditions, that there shall he 
no tension at any horizontal joint , safe unit stresses 
everywhere , and no possible sliding along any plane 
joint , seem to be generally accepted as essential 
to a good design. 

The writer suggests one more condition, that 
the factors of saj'ety against overturning about any 
joint on the outer face shall increase gradually as 
we proceed upwards from the base , to allow for 
the proportionately greater influence, on the 
higher joints, of the effects of wind and wave 
action, ice, floating bodies, dynamite, or other 
accidental forces. The exact amount of increase 
must be largely a matter of judgment; but, if 
the principle is accepted, it can only result in. 
making stronger dams. 


161 



162 


The accompanying sketch of a dam 258 feet 
fiigh to the surface of water (see also “ Engi¬ 
neering News” for June 23, 1888) satisfies the 



described. The dam is of the same total height 
<265 feet) and volume (nearly) as the proposed 
Quaker-Bridge dam, and, for ease of compari- 














163 


son, is designed, as was that dam, for masonry 
weighing 2£ times as much as water. The dam 
is 24 feet wide at top, 38 feet wide 50 feet below 
the surface of water (7 feet below the top), and 
106.1 feet wide at the base. The up-stream 
face is vertical for the first 57 feet from the 
top, and then batters at the rate of 30 feet in 
200 to the base. The outer face slopes uniform¬ 
ly from the top to 50 feet below the water 
surface, and then slopes uniformly to the base. 

The curves of pressure for reservoir, full or 
empty (the lines connecting the centres of 
pressure on the different horizontal joints are 
here styled the curves of pressure), are found as- 
hitherto explained, and are seen to lie well 
within the middle third of the base, so that the 
horizontal joints under the static pressure are 
only subjected to compression throughout their 
whole extent. Further, it was found by con¬ 
struction, that if a horizontal force be assumed 
as acting at the surface of water, of such inten¬ 
sity (29,375 pounds) as to cause the total re¬ 
sultant, on the joint 50 feet below the water 
level, to cut the joint one-third of its width 
from the outer face; then if this same force, 
acting at the surface of water, is combined in 
turn with each of the other resultants on the 
lower horizontal joints, the new centres of press¬ 
ure will still lie well within the middle third 


' % 


1G4 


for the lower joints. 1 To secure uniformity of 
results for all the joints, the width at the 50 
feet level should be increased, although it is 
now much greater than ordinarily constructed. 
If, however, the effects of earthquake vibrations 
are to be guarded against, we cannot replace 
them by the action of a single force acting at 
the surface, so that the increased width of the 
upper joints must be largely a matter of judg¬ 
ment. 

The numbers to the right of the figure, in the 
form of a fraction, give for the corresponding 
joints, for the upper numbers, the factor against 
overturning, or the factor by which it is neces¬ 
sary to multiply the static horizontal thrust of 
the water to cause the total resultant to pass 
through the outer edge of the joint considered ; 


1 It is slated in Engineering News for June 30, 1888, on 
the authority of Mr. Thomas C. Keefer, President American 

Society of Civil Engineers, that “ an ice bridge of about 90 
feet span, between two fixed abutments, expanded so from 
a rise of temperature, as to rise 3 feet in the centre.” If we 
regard the arch thus formed as free to turn at the abutments 
and at the crown, we easily find for ice one foot thick, the 
horizontal thrust II exerted at the abutments, from the 


equation, 3// = 


62. 5 

2 


x 45 2 , to be in pounds per square foot 


11 = 21,094 pounds. Much higher pressures may possibly be 
experienced sometimes near the top of high dams in north¬ 
ern latitudes, and it seems only proper to include such con¬ 
tingencies in their design. 



and for the lower numbers, the ratio of the 
weight of masonry above a joint to the static 
thrust of water against it; which is, in a certain 
sense, a factor of safety against sliding on a 
horizontal joint. These factors are seen to 
increase from the base upwards, so that the 
suggested fourth condition is satisfied. 

The unit stresses, in pounds per square foot, 
at the outer edges of the joints for reservoir 
full, and at the inner edges for reservoir empty, 
are given in columns 4 and 5 of the follow¬ 
ing table, being computed from the formula 

p = ^4 — y of Chap. I. 




Depth of 
Joint below 
Water 
Level. 

1 . 

Water 

Pressure. 

2. 

Weights 

of 

Masonry. 

3. 

Pressure 

at 

outer edge. 

4. 

Pressure 

at 

inner edge. 

5. 

feet. 



lbs. 

lbs. 

50 

1,250 

4,417 

8,860 

10,460 

100 

5,000 

11,540 

13,480 

16,130 

150 

11,250 

23,420 

20,410 

21,440 

200 

20,000 

40,040 

27,330 

27,170 

250 

31,250 

61,420 

34,350 

33,130 

25S 

33,282 

65,270 

35,360 

34,120 


The numbers of columns 2 and 3 for one foot 
in length of the wall are expressed in weights 
of cubic feet of water, and must be multiplied 
by 62.5 to reduce to pounds. 















1GG 


The unit pressures, although necessarily high, 
are still permissible. By spreading the lower 
part of the dam still more, these uuit stresses 
would be theoretically diminished, though it is 
likely that in reality the pressures at the positions 
of the old toes would not be very materially 
altered ; but the masonry being surrounded with 
other masonry could, most probably, stand a 
higher pressure. 

The so-called factors against overturning are 
not true ones, for a computation shows that if 
the water pressure down to the joints 50, 100, 
and 150 feet below the surface should become 
2, H, 1^ times the original, respectively, that 
tension would just begin to be exerted at the 
inner face. This would happen for lower joints 
for thrusts about 1£ to 1£ times the original. 
If, from any cause, as accidental forces at the 
top, earthquakes, etc., the thrusts should be 
increased over these amounts, causing tension 
at the inner edges beyond the capacity of the 
mortar to withstand, the joints would crack and 
open, water would get in, diminishing the weight 
of the masonry materially, the centres of press¬ 
ure would move outwards, and the unit pressures- 
at the outer toes would very much increase,, 
leading perhaps ultimately to the destruction 
of the dam through sliding, overturning, or 
crushing at the down-stream face. 


1G7 


We shall now consider the capacity of resist¬ 
ance of the dam to sliding along any oblique joint 
as AK. 1 Let AB represent, in magnitude and 
direction, the resultant of the water pressure 
and weight of masonry on the horizontal joint 
AH, and let the vertical AD represent the 
weight of the triangular mass A HK, all for one 
foot in length of the wall. Draw DN J _ AK 
and BN jj AK to intersection N; then DN — 
■component of BD normal to plane AK, and 
* DN x tan <j> (where tan (j) = co-efficient of 
friction of masonry on masonry) is the total 
friction that can be exerted by the plane AK. 
If we lay off angle NDE = <p (taken as 35° 
here) to intersection E with the parallel com¬ 
ponent BN, we have DN tan $ = EN, so that 
BE must be resisted by cohesion; and the unit¬ 
shearing stress along the plane AK — 


If, now, we produce KE on to intersection C, 
with AB produced, we have the unit shear rep- 
BC 

resented by —which is a maximum, for 
J AB 


-various planes passing through A, when C is 
farthest removed from B. 

On effecting this construction, then, for a 
series of planes passing through A, we quickly 




1 See Annales des Fonts et Chauss^es for May, 1887. 


' I 







168 


find the plane which will have to supply the 
maximum intensity of shear, or the plane of 
rupture, to lie near AK (there is very little 
difference for a series of planes lying near each 
other) ; and the shear per square foot required 
to resist sliding, in addition to the frictional 
resistance, to be about twenty-seven hundred 
and fifty pounds. To offer the greatest resist¬ 
ance to sliding, there should be no regular 
courses, and the stones should break joint 
vertically as well as horizontally, or the courses 
near the outer face should be curved so as to 
be approximately normal to that face. For a 
retaining-wall of dry rubble, carelessly laid, wo 
see that there is every probability of failure by 
sliding along some inclined plane. Here the 
stones must be carefully interlocked to prevent 
sliding. For the reservoir-wall, where the best 
cement is used, and the joints are broken, there 
should be no fear of sliding when sufficient 
thickness is given to avoid tension. In the 
Habra dam, a hundred and sixteen feet high,, 
this was not done; and the dam broke along a 
plane, passing through the outer toe nearly, and 
making the angle of friction <j> of masonry on. 
masonry with the horizontal. 

It is well to note, too, that friction alone will 
not prevent sliding along planes inclined not 
far from the horizontal as well as those below r 


so that a proper resistance to shear must be 
provided for in every dam. Possibly the weak 
point of many dams is in this very particular. 

The capacity of the dam in question to resist 
rotation about the toe of an inclined base may 
next be tried, and it will be found to be stable; 
for the weight of masonry, as well as its arm, 
increases to counterbalance the increase of arm 
of the water-thrust. The dam thus satisfies all 
the conditions of stability; and, although some 
of its dimensions may be changed with advantage 
perhaps, it yet suffices very well to point out the 
principles of design. 




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